Quantum non-demolition - Quantum Optics and Spectroscopy

Transcription

Quantum non-demolition measurements
and quantum simulation
Thesis submitted to the
Faculty of Mathematics, Computer Science and Physics
of the Leopold-Franzens University of Innsbruck
in partial fulfillment
of the requirements for the degree of
Doctor rerum naturalium
by
Gerhard Kirchmair
Innsbruck, July 2010
2
Abstract
Quantum information processing (QIP) has developed into one of the “hot topics” in physics during
the last decade. More and more experiments appeared aiming at the realization of a quantum
computer and/or quantum simulations. Strings of ions stored in a Paul trap are currently one of
the most promising systems for realizing quantum algorithms. The great challenges ion traps are
facing is the realization of robust high fidelity operations, scalable traps and the combination of
all DiVincenzo criteria in one system.
This thesis reports on the realization of simple QIP sequences and a quantum simulation with
trapped 40 Ca+ and 43 Ca+ ions. For both isotopes the quantum bit (qubit) is encoded in an optical
transition. It consists of the ground state S1/2 and the metastable excited state D5/2 . The isotope
43
Ca+ has a nuclear spin of I = 7/2 and thus exhibits a very complicated level structure. The
advantage of this isotope is that hyperfine levels can be used as a robust memory for quantum
information.
In the framework of this thesis a high fidelity Mølmer-Sørensen gate operation was implemented.
Two ion Bell states and three ion GHZ states were created with a respective fidelity of 99.3% and
98.4% using 40 Ca+ ions. It was demonstrated that the Mølmer-Sørensen gate operation works
nearly as well for ions cooled to the ground state as for ions in thermal states of motion. The
fidelity achieved for thermal ions was still 98.4%. A thorough experimental analysis of the gate
mechanism indicated possible error sources.
A combination of four DiVincenzo criteria (initialization, long coherence times, a universal set of
gates, readout) was realized by utilizing 43 Ca+ hyperfine states. Bell states, created on the optical
qubit, were stored in the qubit S1/2 , F = 4, mf = 0 ↔ S1/2 , F = 3, mf = 0 for 20 ms.
The experience gained with the entangling operation rendered it possible to do non-demolition
two-qubit measurements. The non-demolition measurements were used to experimentally test
hidden variable theories, more precisely the Kochen Specker theorem. The experiments conducted
showed that hidden variables assuming non-contextuality of measurements cannot reproduce the
behavior of quantum systems. This was proven by violating an inequality akin to Bell tests. The
experimental results also demonstrated the state-independence of the Kochen Specker argument.
A possible compatibility loophole was closed by taking into account experimental errors in the
theory. The resulting modified equation was found to be violated in our experiments.
Finally, a proof of principle quantum simulation of the Dirac equation was implemented. Position
and momentum of the Dirac particle were mapped onto the respective quadrature components of
the ion trap harmonic oscillator. A new method to measure the expectation value of position and
momentum was implemented. The particle trajectories obtained showed effects like Zitterbewegung.
Using the high degree of control available in the experiment specific initial states were created
showing different trajectories.
i
Abstract
ii
Zusammenfassung
Quanteninformationsverarbeitung (QIV) hat sich in eines der meist diskutierten Themen in der
Physik im letzten Jahrzehnt entwickelt. Immer mehr Experimente werden gebaut, die darauf abzielen, einen Quantencomputer oder Quantensimulator zu realisieren. Besonders Ionen, gespeichert
in einer Paul-Falle, gehören zu den vielversprechendsten Systemen, um Quanten-Algorithmen zu
realisieren. Die großen Herausforderungen, die es bei Ionenfallen zu lösen gilt, sind die Realisierung
von Quantengattern mit hoher Güte, der Bau von skalierbaren Fallen und die Kombination aller
DiVincenzo Kriterien in einem System.
Die vorliegende Arbeit beschreibt die Realisierung von einfachen Sequenzen zur QIV und zu
Quanten-simulation mit 40 Ca+ und 43 Ca+ Ionen. Bei beiden Isotopen erfolgt die Kodierung des
Quanten-Bits (Qubit) in einem sogenannten optischen Qubit, bestehend aus dem Grundzustand
S1/2 und dem metastabilen angeregten Zustand D5/2 . Das Isotop 43 Ca+ hat einen Kernspin von
I=7/2 und daher ein sehr kompliziertes Termschema. Der Vorteil dieses Isotops ist, dass Hyperfein
Niveaus, sogenannte ”Uhrenübergänge”, als robuste Speicher für Quanteninformation verwendet
werden können.
Im Rahmen dieser Doktorarbeit wurde ein Mølmer-Sørensen Gatter hoher Güte realisiert. Es
wurden mit diesem Gatter Bell-Zustände mit eine Güte von 99.3% und GHZ-Zustände, bestehend
aus drei Ionen, mit einer Güte von 98.4% hergestellt. Weiters wurde gezeigt, dass das MølmerSørensen Gatter auch mit Ionen funktioniert, die nicht in den Grundzustand gekühlt wurden. Die
erreichte Güte unter diesen Voraussetzungen betrug immer noch 98,4%. Mögliche Fehlerquellen
der Gatter Operation wurden durch eine gründliche Untersuchung quantifiziert.
Durch Verwendung von 43 Ca+ Hyperfein-Zuständen war es möglich, vier der fünf DiVincenzo
Kriterien (Initialisierung, lange Kohärenzzeiten, universeller Satz an Gattern, Zustands- Detektion)
in einem System zu vereinen. Bell Zustände, die auf dem optischen Qubit erzeugt wurden, konnten
ins Hyperfein-Qubit S1/2 , F = 4, mF = 0 ↔ S1/2 , F = 3, mF = 0 übertragen werden und für mehr
als 20 ms gespeichert werden.
Die mit den Gatter Operationen gewonnene Erfahrung ermöglichte die Realisierung von nicht destruktiven Quantenmessungen. Diese nicht-destruktiven Messungen wurden dazu verwendet um sogenannte Theorien versteckter Variablen zu testen. Die durchgeführten Experimente zeigten, dass
Theorien versteckter Variablen, aufbauend auf nicht kontextuellen Messungen, quantenmechanische Systeme nicht richtig beschreiben. Dies wurde gezeigt durch Verletzung einer Ungleichung sehr
ähnlich der Bellschen Ungleichung. Weiters ergaben die Messungen, dass die Verletzung der Ungleichung unabhängig vom Quantenzustand ist. Das Kompatibilitätsschlupfloch wurde geschlossen
durch Einbeziehen der experimentellen Fehler in die Theorie.
Letztendlich wurde eine Quantensimulation der Dirac Gleichung experimentell realisiert. Ort
und Impuls des simulierten Teilchens wurden auf die betreffende Quadraturkomponente des har-
iii
Zusammenfassung
monischen Oszillators abgebildet. Eine neue Methode wurde verwendet, um den Erwartungswert
des Orts und Impuls des simulierten Teilchens zu bestimmen. Die gemessene Teilchentrajektorien
zeigten relativistische Effekte wie Zitterbewegung. Durch Ausnützen der außergewöhnlichen experimentellen Kontrolle konnten verschiedene Anfangszustände erzeugt werden die unterschiedliche
Trajektorien zeigten.
Contents
Abstract
i
Zusammenfassung
iii
1. Introduction
1
2. Trapped calcium ions as qubits
5
2.1. Quantum computation and quantum bits . . . . . . . . . . . . . . . . . . . . . . .
5
2.2. Quantum harmonic oscillator and phase space . . . . . . . . . . . . . . . . . . . . .
10
2.3. Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3. Laser-ion interaction
19
3.1. Ion laser Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.2. Non-resonant interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.3. The S1/2 → D5/2 transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.4. Microwave transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.5. Mølmer-Sørensen gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.6. Creating a coherent displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
4. Experimental setup
33
4.1. Linear ion trap and electrode wiring . . . . . . . . . . . . . . . . . . . . . . . . . .
33
4.2. Vacuum vessel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
4.3. Magnetic field coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
4.4. Optical access and single ion addressing . . . . . . . . . . . . . . . . . . . . . . . .
36
4.5. Fluorescence detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
4.6. Laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
4.7. Computer control and RF generation . . . . . . . . . . . . . . . . . . . . . . . . . .
45
5. Experimental techniques
47
5.1. Loading ions by photoionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
5.2. Experimental sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
5.3. Referencing the 729 nm laser to the ions . . . . . . . . . . . . . . . . . . . . . . . .
51
5.4. Implementation of arbitrary qubit operations and ion shuttling . . . . . . . . . . .
52
5.5. State tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
Contents
6. Entangling calcium ions
6.1. High fidelity two ion Bell states . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2. Three ion entangled states and arbitrary operations on three qubits . . . . . . . .
57
57
67
6.3. High fidelity entanglement of 43 Ca+ hyperfine clock states . . . . . . . . . . . . . .
6.4. Comparison between 40 Ca+ and 43 Ca+ . . . . . . . . . . . . . . . . . . . . . . . . .
70
74
7. Experimental test of quantum contextuality
7.1. The Kochen-Specker theorem . . . . . . . . . . . . .
7.2. Experimental realization of QND measurements . . .
7.3. Testing the Kochen Specker inequality . . . . . . . .
7.4. Including imperfect measurements to close loopholes
.
.
.
.
77
78
81
83
86
7.5. Experimental results on imperfect measurements . . . . . . . . . . . . . . . . . . .
89
8. Quantum simulation of the Dirac equation
8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2. The Dirac equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3. Experimental realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
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94
9. Summary and Outlook
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101
A. The Kochen Specker measurements
105
A.1. Detailed results for the Kochen Specker measurements . . . . . . . . . . . . . . . . 105
A.2. Hiding the second ion in the Kochen Specker measurements . . . . . . . . . . . . . 106
B. Methods for simulating the Dirac equation
107
B.1. Constructing a pure negative energy spinor . . . . . . . . . . . . . . . . . . . . . . 107
B.2. Setting the phase of the Displacement pulse . . . . . . . . . . . . . . . . . . . . . . 107
C. Physical and optical properties of Calcium
109
D. Journal Publications
111
1. Introduction
Quantum mechanics (QM) was among the greatest developments in the twentieth century. The
foundations of this theory were established around 1925 by Werner Heisenberg, Erwin Schrödinger,
Max Born, Wolfgang Pauli, Niels Bohr, Albert Einstein, Paul Dirac and many others. Initially
invented for a better understanding of atomic spectra, quantum mechanics opened the door to
a completely new field in physics. Nowadays it successfully explains chemical processes, material
behavior and semiconductor devices, enabling the development of the modern computer technology.
Computers in turn helped to numerically solve quantum mechanical problems which could not be
solved analytically. In the following decades physicists realized that this approach has its limits as
quantum systems consisting of only a few quantum bits (qubits) are already impossible to simulate
with current computer technology.
In the early 1980’s Paul Benioff [1] and Richard Feynman [2] came up with the idea of simulating
one quantum system by using another one. This so called quantum computer would only need an
amount of qubits on the same order as the simulated system. Back in the 80’s these ideas where
considered as mere “Gedankenexperimente“ and not as realistic scenarios that could be developed
into experiments. It was considered impossible to use a single atom as a carrier for quantum
information. Richard Feynman said in a lecture in 1986: ”. . . we are to be even more ridiculous
later and consider bits written on one atom instead of the present 1011 atoms. Such nonsense is
very entertaining to professors like me. I hope you will find it interesting and entertaining also.“
Today such procedures are experimental routine and quantum information processing as well
as quantum simulation are major research fields in theoretical and experimental physics. The big
interest was sparked by the discovery of a factorization algorithm on a quantum computer, which
is much faster than its classical counterpart, by Peter Shor in 1994 [3]. A large scale experimental
realization of this algorithm would render public key encryption useless. Two years later Lov
K. Grover found that a quantum computer can search unsorted databases faster than a classical
computer [4]. Barenco et. al. [5] as well as DiVincenzo [6] showed in 1995 that an arbitrary
quantum algorithm can be constructed with a limited set of single and two-qubit gates. This
“universal“ set of operations is similar to the classical approach of constructing logic circuits out
of NAND gates. To actually build a quantum computer certain prerequisites have to be met.
These requirements were formulated by DiVincenzo [7] and are often called DiVincenzo criteria,
according to which any quantum computer implementation requires
1. A scalable physical system with well characterized qubits
2. The ability to faithfully initialize the qubit to a simple well defined state
3. Long relevant coherence time, much longer than the gate operation time
4. A “universal“ set of quantum gates
1
Introduction
5. Qubit specific measurement capability
Later two additional requirements were added to incorporate quantum communication between
distant nodes of a quantum network
6. The ability to interconvert stationary and flying qubits
7. The ability to faithfully transmit flying qubits between specified locations
Throughout the last decade research has shown thtat there are several experimental implementations that fulfill most or even all of these points and might be considered as candidates for a quantum computer: nuclear spins [8], superconducting Josephson junctions [9, 10], quantum dots [11],
neutral atoms [12], photons [13] and ions [14]. While nuclear spins are the most advanced system
when it comes to an actual implementation of algorithms and the number of qubits [15], they suffer
from the inability to create pure states. This prevents scaling of this technology to higher numbers
of quantum bits. Solid state implementations offer the promise of easy scalability akin to integrated
circuits as soon as one is able to reliably manufacture and control the basic building blocks. In the
last few years these devices made remarkable progress realizing Bell states [16], two-qubit gates [17]
and quantum process tomography [18]. The limiting factor at the moment is the coherence time of
a few microseconds [19]. Recent experiments with neutral atoms demonstrated the extraordinary
control physicists have over single atoms [20] and ultracold ensembles [21]. These systems seem to
be good candidates for quantum simulations of solid state models [22].
One of the most promising systems for realizing a quantum computer are trapped ions [23]. They
already fulfilled three of the five DiVincenzo criteria even before they were considered as a system
for quantum information processing (QIP): initialization [24], readout [25–27] and long coherence
times [28]. Laser-cooled strings of ions had been observed[29–31] qualifying as well characterized
qubits which could be used as a quantum register. The trapping of charged particles in electric
fields was first proposed by K.H. Kingdon in 1923 [32]. Radio frequency traps were developed by
Wolfgang Paul in the 1950’s and a single ion was trapped the first time by Neuhauser et al. in
1980. The trapping of ions had a huge influence on atomic physics and sparked the development
of optical clocks and mass spectrometers. Furthermore it opened up the possibility of controlling
well-isolated quantum systems for processing quantum information. This point was first noticed
in the seminal paper by Peter Zoller and Ignacio Cirac in 1995 [14]. They proposed to use the
collective motion of an ion string as a bus system to couple individual ions. A series of laser pulses
acting on one ion at a time can be used to realize a two-qubit gate. This approach is also scalable
as the resources increase only polynomially with the number of qubits. Within the same year a
two-qubit gate was demonstrated [33] with a single ion which set the starting point for quantum
information processing with trapped ions.
In the last years ion trap experiments have shown a tremendous control of experimental parameters. This control was the key to demonstrate the first steps towards a quantum computer and/or
quantum simulations. Meanwhile all seven DiVincenzo criteria have been achieved in at least
proof of principle experiments. Some of the milestones are the investigation of several two-qubit
gates [34–40], deterministic quantum teleportation with atoms [41, 42], quantum error correction [43], entanglement purification [44], the creation of six [45] and eight ion entangled states [46]
2
and the implementation of simple algorithms like the Deutsch-Josza algorithm [47] and Grovers
search algorithm [48]. Along the way, several analysis techniques were developed like quantum
state tomography and quantum process tomography [49, 50]. The so called decoherence-free subspace was investigated [51] and universal ion-trap quantum computation with decoherence-free
qubits was realized [52]. Ions trapped in separate traps have been entangled [53] and quantum
teleportation has been demonstrated [54] fulfilling the additional DiVincenzo criteria.
Today the big challenges on the road towards an ion trap quantum computer are the following:
scaling current technology to a few ten or hundred qubits; combining all building blocks in one
system; improving all operations to enter the fault tolerant regime. Scaling the trap technology
to a high number of ions is a difficult but important task as for a real commercially useful implementation of Shor’s factoring algorithm some 105 qubits are needed. Several research groups are
currently constructing and testing miniaturized traps but progress is slow. Recent experiments
have demonstrated the combination of the basic DiVincenzo criteria in one experiment [55, 56].
Entering the fault tolerant regime is important for realizing a large scale quantum computer
capable of doing arbitrary long quantum algorithms. For this purpose quantum error correction
protocols similar to classical computation are indispensable. The first protocols were discovered
by Peter Shor [57] and Andrew Steane [58]. Furthermore, Peter Shor showed that it is possible to
realize quantum algorithms with arbitrarily small errors even when using imperfect operations [59].
This is possible as soon as all operations enter the fault tolerant regime which means errors below
10−2 . . . 10−4 . The exact threshold depends on the overhead in the number of qubits one is willing
to accept [60–62].
Even though all building blocks for an ion trap quantum computer have been demonstrated it
will still take time until a large scale functioning quantum computer will become reality. Nevertheless, ion trap systems might soon outperform classical computers when it comes to quantum
simulations. Here the demands on the quality of the operations and the number of qubits is much
less. Decoherence might be even used to mimic the effect of an environment. Recently a proof
of principle experiment showed the simulation of a quantum magnet [63], demonstrating a phase
transition from a paramagnetic to a ferromagnetic order with two-qubits. Another proof of principle experiment, the simulation of the free Dirac equation will be presented in chapter 8. So even
relativistic quantum effects might be simulated with a quantum computer. Soon experiments will
be able to control and do simulations with 10-20 interacting qubits which is already a next to
impossible problem for a classical computer.
The high control over experimental parameters in ion traps opens up the possibility to do
fundamental tests of QM. These test are important as it is still debated whether quantum mechanics
can be explained with so called hidden variable models. Hidden variables were introduced as some
physicists were unhappy with the Copenhagen interpretation of quantum mechanics which only
predicts the probabilities for the outcome of a measurement. Another intriguing attribute of QM
are its nonlocal features, e.g. a measurement of two entangled particles shows correlations which are
bigger than classically possible. This was first discovered by Einstein, Podolsky and Rosen in their
famous paper from 1935 [64]. This point lead physicists to the conclusion that QM is incomplete
and other unknown physical phenomena beyond QM have to be included. In 1964 John Bell showed
that hidden variable theories have to be non local to explain all effects of quantum mechanics [65].
3
Introduction
He presented his famous Bell inequality which is satisfied by local hidden variables but violated
by QM. Several experiments have meanwhile proven that QM violates Bell’s inequality [66–69].
Another class of hidden variable theories are non-contextual theories which are the subject of a
theorem developed by Kochen, Specker and Bell [70–72] in the 60’s. Recent experiments [73–75],
one of them presented in Chapter 7, have shown that also this model cannot explain QM. So far
it seems there is no better theory than QM covering all observed phenomena.
The thesis is structured as follows: Chapter 2 contains a review of the main ideas and terms of
quantum information processing followed by a quick review of the quantum harmonic oscillator.
The level structure of 40 Ca+ and 43 Ca+ ions is described and possible choices for qubits are illustrated. Chapter 3 summarizes the interaction of the ions with electromagnetic fields with a special
focus on the Mølmer-Sørensen entangling gate. Chapter 4 describes the apparatus consisting of the
vacuum chamber, the lasers and the electronic control. Chapter 5 explains standard experimental
techniques necessary for running the apparatus and realizing the experiments presented in the subsequent chapters. Chapter 6 describes the realization of an entangling operation working on 40 Ca+
and 43 Ca+ ions. Bell states with fidelities as high as 99.3% were created and the gate operation
was analyzed in detail. Furthermore we achieved an entanglement of 40 Ca+ ions in thermal states
of motion and a mapping of the entanglement from the optical qubit onto the hyperfine qubit
in 43 Ca+ Ḟinally three-ion entangled states with 40 Ca+ were created and the first block of a new
scheme for universal quantum computation with trapped ions was implemented. Chapter 7 details
on a test of the non-contextual hidden variable theory proposed by Kochen, Specker and Bell in a
two ion system. Data are presented that demonstrate that QM measurements are contextual even
in the presence of experimental noise. Chapter 8 describes the realization of an experiment with
a single ion simulating the free Dirac equation. The particle position measured as a function of
time shows Zitterbewegung and other relativistic effects. Chapter 9 concludes the thesis with a
summary and outlook to future experiments.
The main findings of this thesis were published in the references [75–78].
4
2. Trapped calcium ions as qubits
This chapter will quickly review the mathematical notation for a two-level system and the basic
ideas of quantum computation. The mathematical excursus is continued by a description of a quantum harmonic oscillator with an emphasis on canonical variables and the phase space description.
The energy level structure of the Calcium isotopes 40 Ca+ 43 Ca+ will be outlined and a number of
possible implementations of a qubit in calcium ions will be shown.
2.1. Quantum computation and quantum bits
Inspired by [79] this section will present the mathematical description of a two-level system and
will give a brief summary on quantum computation. It will introduce the notions “qubit“ and
“entanglement“ as well as single and two-qubit operations.
2.1.1. Qubits
For a real physical implementation of a quantum computer, quantum information has to be stored
and manipulated. Akin to classical computation we will make use of two states of a system. These
states will represent the quantum states |↓i and |↑i instead of the logical 0 and 1. In trapped ions,
a qubit can be realized by identifying two long-lived atomics states with the qubit states |↓i and
|↑i. The main difference from the classical case is that the system cannot only be in one of the
two states but also in a linear superposition
|Ψi = α |↓i + β |↑i
(2.1)
with α and β complex numbers and the normalization condition |α|2 + |β|2 = 1. Such a two-level
system is called quantum bit/qubit in analogy to the classical case. The numbers |α|2 and |β|2
represent the probability of measuring the qubit in state |↓i or |↑i. A single measurement will just
show either |↓i or |↑i. Only by many repetitions the full information about a quantum state can
be acquired.
A convenient and often used form of rewriting equation (2.1) is
µ ¶
µ ¶
θ
θ
|↓i + sin
eiφ |↑i
|Ψi = cos
2
2
(2.2)
where a global phase factor was dropped that is not observable in experiments. Thus a pure
quantum state is completely described by two real numbers θ and φ. This representation allows us
to visualize the qubit’s state on sphere a with unit radius, where θ and φ are the rotation angles
of the Bloch vector. This sphere, called Bloch sphere, is shown in figure 2.1.
5
Trapped calcium ions as qubits
Figure 2.1.: Bloch sphere representation of a pure qubit state |Ψi described by the rotation
angles θ and φ.
The state description can be extended to N-qubits by using a set of basis states composed of the
2 product states
|ni = |iN i ⊗ . . . ⊗ |i2 i ⊗ |i1 i
(2.3)
N
with i ² {0, 1}. With the vectors |ni we can now describe every possible state consisting of N qubits
as
N −1
2X
ΨN =
αn |ni
(2.4)
n=0
P2N −1
2
where the state amplitudes αn have to satisfy the normalization condition
n=0 |αn | = 1.
Unfortunately, the convenient Bloch sphere representation is not extendable to more than one
qubit.
2.1.2. Quantum gates
A universal language to describe quantum algorithms is the quantum circuit model. In analogy
to classical computing, where a computation is built up of logic gates, in this model a quantum
computation consists of concatenations of elementary quantum operations acting on the qubits.
It can be shown, that there exists a small set of single qubit operations and a universal two-qubit
gate that can be used to perform arbitrary quantum algorithms [79].
The simplest interaction acting on an N-qubit system is a single qubit operation. These operations have to preserve the norm of the state and can be conveniently described by using the Pauli
matrices plus the Identity matrix
Ã
σx =
0
1
1
0
!
Ã
, σy =
0
−i
i
0
!
Ã
, σz =
1
0
0
−1
!
Ã
, Iˆ =
1
0
0
1
!
.
(2.5)
In the Bloch sphere representation, single qubit operations can be described as a rotation around
the x, y, z axis or an arbitrary axis φ in the equatorial plane by an angle θ. The rotation matrixes
6
2.1 Quantum computation and quantum bits
are given by
Ã
=e
−i θ2 σx
=e
−i θ2 σy
=e
−i θ2 σz
(θ, φ) = e
−i θ2 σφ
Ux(j) (θ)
=
Ã
Uy(j) (θ)
=
Ã
Uz(j) (θ)
=
Ã
U
(j)
=
cos θ2
−i sin θ2
!
cos θ2
!
− sin θ2
cos θ2
!
0
θ
ei 2
(2.6)
−i sin θ2
cos θ2
sin θ2
θ
e−i 2
0
−ie−iφ sin θ2
cos θ2
cos θ2
−ieiφ sin θ2
(2.7)
(2.8)
!
,
(2.9)
where σφ = cos(φ) σx + sin(φ) σy and j denote the ion subject to the rotation. Setting φ = 0
(i)
(i)
in U (i) (θ, φ) results in a Ux (θ) rotation and similarly setting φ = π/2 results in Uy (θ). If we
can apply these rotations to every qubit of our quantum computer it is sufficient to introduce one
particular type of multi-qubit operation to construct all arbitrary operations. There are many
different choices for universal gate operations. The best known example is the controlled -NOT
(CNOT) operation where the state of the target qubit is flipped or not depending on the state of
the control qubit. The matrix representation of this operation is

UCN OT


=


1
0
0
0
0
1
0
0
0
0
0
1
0
0
1
0






(2.10)
where the matrix is notated with respect to the basis order {|↓↓i , |↓↑i , |↑↓i , |↑↑i}. Another important two-qubit operation is the Mølmer-Sørensen [80, 81] (MS) gate operation. This gate was
implemented in this thesis and will be further discussed theoretically in chapter 3 and experimentally in chapter 6. Its matrix representation is

UxM S
³π ´
2
π
= e−i 4 σx ⊗σx =

1−i

2 


1 0
0 i

0 1 −i 0 
.
0 −i 1 0 

i 0
0 1
(2.11)
This gate introduces collective spin flips on two ions and can be extended to an N-qubit gate
(see chapter 3.5). The normalization factor in front of the matrix is required because two-qubit
†
ˆ The Mølmer-Sørensen gate is
operations have to preserve the norm such that Ugate
Ugate = I.
completely equivalent to a CNOT operation up to some single qubit rotations.
2.1.3. Quantum state measurement
At the end of a quantum algorithm/simulation during which the system evolved according to a
unitary operation the outcome needs to be determined. This requires a detection of the qubit states
by a projective measurement. Its experimental implementation for the case of trapped ions will be
7
discussed in chapter 4.5. Mathematically this process can be described by a set of measurement
operators {Mm } such that the results m of a measurement occur with a probability
†
p(m) = hΨ| Mm
Mm |Ψi .
(2.12)
The state of the system after the measurement is one of the eigenstates of the measurement operator
¯ E
¯
¯Ψ̃ = q
Mm |Ψi
(2.13)
†
hΨ| Mm
Mm |Ψi
The natural measurement basis for trapped ions are the projectors onto the energy eigenstates.
Thus one can only measure the σz components of every qubit state Ψ by a simple fluorescence
detection. If projections onto other spin components A = ~n~σ are of interest (where ~n is the
direction of the desired spin component) a modified technique has to be employed.
The measurement basis can be changed by applying a unitary state transformation U prior to
+
the state detection on each qubit individually. This means the subspace HA
= {ψ|Aψ = ψ} is
+
0
0
0
mapped onto the subspace Hσz = {ψ |σz ψ = ψ }. Thus a σz measurement on the transformed
state |Ψ0 i = U |Ψi is equivalent to a measurement of the desired observable A on the original state
|Ψi. The unitary state transformation U satisfies
A = U † σzi U
(2.14)
and is given by a rotation around an axis on the Bloch sphere that is both perpendicular to ~ez and
to ~n. The final inverse operation U † maps the projected state onto an eigenstate of the observable
A which is in general not necessary. Nevertheless it becomes important when one is interested in
doing non-demolition measurements in succession (see chapter 7). The mapping on an eigenstate
ensures that subsequent commuting measurements will still determine the right value.
An additional complication appears if e.g. a spin correlation like σz ⊗ σz , the so called parity,
of a two ion state should be measured. If exactly this observable should be determined in a nondemolition measurement it is necessary to apply entangling two-qubit gates together with single
qubit operations, that map this information on a single qubit. An example for such an operation
U is a CNOT gate which will map the parity of the two ions on the state of the target ion.
initial state
mapped state
inital state
σz ⊗ σz
mapped state
I ⊗ σz
|↓↓i
|↓↑i
(-1) · (-1) = +1
+1
|↓↑i
|↑↓i
|↓↓i
|↑↓i
(-1) · 1 = -1
1 · (-1) = -1
-1
-1
|↑↑i
|↑↑i
1 · 1 = +1
+1
By measuring only the state of the second ion after the mapping, it is ensured, that the parity
measurement reveals a single bit of information. It is important to note, that such a measurement is
not equivalent to measuring σz on every ion and correlating the results afterwards to get σz ⊗ σz .
In the latter case one would not only get the information whether the ions are in the same or
opposite spin state but also in which state each ion is.
8
2.1 Quantum computation and quantum bits
2.1.4. Entanglement
Called Spooky Action at a Distance [64] by Einstein, Podolsky and Rosen, entanglement brings up
some of the most striking features of quantum mechanics. Let us consider an entangled state for
two qubits, the so called Bell state
|↑↑i + |↓↓i
√
|Ψi =
(2.15)
2
which was created in several experiments described in this thesis (see chapter 6). Whenever the
state of one qubit of this entangled state is measured, one will find a perfectly correlated result
when measuring the other. If the first qubit is in the state |↓i /(|↑i) the second qubit will be in
|↓i /(|↑i) or vice versa so both qubits will always be found in the same state. This behavior will
violate one or both of the assumptions Einstein and Co. made for a physical realistic theory:
• Realism: The assumption that the physical properties of a system have definite values which
exist independent of observation
• Locality: The assumption that a measurement on one particle does not influence the result
of the measurement on the second particle.
These assumptions seem to be intuitively plausible but nevertheless experiments conducted with
these states [66] show that they disobey the so called Bell inequality [72]. This then leads to
the conclusion that they have to disobey one or both of the upper points. There is still a lot of
discussion going on about the results of these experiments that go beyond the scope of this work.
More information about this subject can be found in references [82–84].
A more rigorous criterion for entangled states can be phrased as: A state is entangled if its
composite wavefunction cannot be written as a product of wavefunctions of the subsystems. Mathematically this means
|ΨAB i 6= |ΨA i ⊗ |ΨB i .
(2.16)
Creating entangled states is nowadays a standard procedure which can be employed in several
different physical systems [46, 85–87]. A single application of a Mølmer-Sørensen-gate to |↓↓i
creates already an entangled state of the form shown in equation (2.15) They can be used as
resources for secure quantum communication [88] or certain types of quantum computation [89].
Entanglement seems to appear in most quantum algorithms however it is still a matter of debate
if the power of an algorithm comes from entanglement or not.
In chapter 7 of this thesis another peculiar feature of quantum-mechanics, the so called contextualtity of measurements, will be discussed. Contextualtity of measurements means that certain
types of measurements influence another even though they should not. The states measured inherit
entanglement not from the beginning but entanglement will be created during the measurement
process. Entanglement plays a central role in quantum mechanics.
2.1.5. Density matrix description
So far the description of a qubit is sufficient for pure states but fails for states which are mixed
or not completely known. Here the density operator language provides a convenient means to
describe these systems. All postulates for quantum mechanics regarding state description, state
9
evolution and measurement can be reformulated in terms of density operators [79]. For a system
which can be in pure states Ψi with a probability pi the density operator is given by
ρ=
X
pi |Ψi i hΨi | .
(2.17)
i
One has to note that the probabilities pi are only well defined for an ensemble of states. A density
matrix has to satisfy the conditions
1. the trace of ρ is equal to one
2. ρ is a positive operator.
For a d-dimensional quantum system it is possible to expand the density matrix ρ using a basis
of d2 hermitian matrices that are mutually orthogonal. A handy choice for a single qubit are the
Pauli matrices together with the identity
ρ=
1
2
X
λi σi .
(2.18)
i=x,y,z,Iˆ
The parameters λi which completely characterize the state are given by
λi = T r{ρσi }.
(2.19)
So each of these parameters corresponds to the outcome of a projective measurement determining
the expectation values of the Pauli operators hσi i. This will become handy for reconstructing the
quantum state in the experiment with a technique called state tomography (chapter 5). As one can
see from equation (2.20) it is sufficient to determine all λi to get the full knowledge of a quantum
state. The parameters λi uniquely define a vector on the Bloch sphere (similar to figure 2.1) but
not only the angles but also the length of the vector is now of importance. The length of the vector
P
is given by x,y,z λ2i and it shows the purity of a state. A pure state has a unit length vector, a
partially mixed state has a vector with a length smaller than 1 and a completely mixed state has
a vector with a length equal to zero.
This description can be straightforwardly extended to multi-qubit systems. The density matrix
for a N-qubit system is written in terms of tensor products of the Pauli matrices
ρN =
1
2N
X
λi1 ,i2 ...iN σi1 ⊗ σi2 ⊗ . . . ⊗ σin .
(2.20)
i1 ,i2 ...iN ,=x,y,z,Iˆ
Normalization requires that λI,I...I = 1 such that 4N −1 parameters define the state in the Hilbertspace. Each mi1 ,i2 ...iN corresponds again to a projective measurement determining hσi1 ⊗ . . . ⊗ σin i.
2.2. Quantum harmonic oscillator and phase space
An ion in a trap is one of the best model systems for a perfect quantum harmonic oscillator.
Usually the harmonic oscillator is only used to mediate interactions between the ions but is not
10
2.2 Quantum harmonic oscillator and phase space
regarded by itself as an interesting system. In chapter 8 it will be shown that using a single ion
in the harmonic oscillator of the trap one can conduct a simulation of the Dirac Equation. The
information of the position and the momentum of the Dirac particle will be encoded in the position
and momentum observables of the harmonic oscillator. Hence this section will reviwe the analysis
of a quantum harmonic oscillator with an emphasis on the phase space description. At the end a
classical driving force will be introduced and the effects on the state of the oscillator will be shown.
This chapter is strongly inspired by [37, 90, 91].
2.2.1. Harmonic oscillator hamiltonian
A quantum harmonic oscillator is a particle of mass m moving in a 1-d quadratic potential V (xc ) =
mω 2 x2c /2. Classically the state of a harmonic oscillator is defined by its position and momentum
coordinates xc and pc . A classical harmonic oscillator is best described in phase space which is
given by the complex plane spanned by the variable xc + ipc . The motion of the particle can be
described by a point in phase space which is rotating with an angular frequency ω along an ellipse
centered at the origin. In quantum mechanics the position and momentum will be replaced by the
observables x̂ and p̂. They obey the commutation relation
[x̂, p̂] = i~.
(2.21)
With these operators the Hamiltonian for a quantum harmonic oscillator reads as
Hm =
p̂2
1
+ mω 2 x̂2 .
2m 2
(2.22)
Introducing the non-hermitian operator a and its conjugate a† such that
x̂ = (a + a† )∆
and
p̂ = i
a† − a ~
2 ∆
(2.23)
p
with the characteristic length ∆ = ~/(2mω) one can rewrite equation (2.22) and arrive at the
well known quantized version of the harmonic oscillator Hamiltonian
µ
¶
1
Hm = ~ω a† a +
.
2
(2.24)
The spectrum of this Hamiltonian is a ladder of equidistant energy levels EN with a spacing of
~ω starting at the ground state energy of E0 = ~ω/2. The corresponding wavefunctions |N i are
called Fock States and will obey the following relations
a† |ni =
a |ni =
√
√
n + 1 |n + 1i
(2.25)
n |n − 1i
(2.26)
11
b
a
Im{a}
Im{a}
D
|a|
h
2D
wt
Re{a}
Re{a}
Figure 2.2.: (a)Phase space representation of a coherent state |αi. The coordinates of the
center of the disk correspond to hxi and hpi. The size of the disk represents the variance of
the state here with unit length along p and x. (b) Time evolution of a coherent state. The
state rotates with the trap frequency ω around the origin.
and
a†n |0i =
√
n! |ni
a† a |ni = n |ni .
(2.27)
(2.28)
Often a and a† are called lowering and raising operators, because they can be used to climb up
and down the ladder of states. The spatial extension of the ground state |0i is given exactly by ∆.
2.2.2. Driven quantum harmonic oscillator
If a harmonic oscillator is driven by a classical force
F (t) = A sin(ωdrive t + φ) =
A i(ωdrive t+φ)
(e
− e−i(ωdrive t+φ) )
2i
(2.29)
the Hamiltonian is given by
Hf orce = −x̂F (t).
(2.30)
The action of this force on the ground state of the harmonic oscillator can be best explained by
changing into the interaction picture. A suitable transformation to do this is given by U |Ψi = |Ψ0 i
with U = eiHm t/~ . The Hamiltonian in the interaction picture then reads as
Hfi orce = −∆(ae−iωt + a† eiωt )F (t).
(2.31)
For ω = ωdrive a resonance appears and using the rotating wave approximation the Hamiltonian
reduces to
A∆
Hfi orce = −
(ae−iφ + a† eiφ )
(2.32)
2i
The evolution of the ground state under the action of this Hamiltonian is given by the unitary
operator
D(α) = eα
12
∗
a−αa†
(2.33)
2.3 Atomic Structure
with α = −A∆t/(2~)e−iφ . This propagator is a so called displacement operator. Applying it to
the ground state of motion creates the coherent state |αi
D(α) |0i = |αi = e−|α
2
|/2
X αn
√ |ni .
n!
n
(2.34)
This can be generalized to cover the displacement of a coherent state |βi by using the Baker–
∗
Campbell–Hausdorff relation D(α)D(β) = D(α + β)ei Im{αβ }
D(α) |βi = D(α)D(β) |0i = ei
Im{αβ ∗ }
|α + β > .
(2.35)
A coherent state obeys the following relations
a |αi = α |αi
(2.36)
hα| a† = α∗ hα|
(2.37)
hα| x̂ |αi = 2∆ Re{α}
(2.38)
~
Im{α}.
∆
(2.39)
and
hα| p̂ |αi =
The real part of α corresponds to the mean position while the imaginary part corresponds to
the mean momentum. A convenient way to display these states is a phase space diagram (see
figure 2.2(a)) with α = hx̂i + i hp̂i with x in units of 2∆ and p in units of ~/∆. The variance for
position and momentum is also given by ∆ as
δx = ∆
and
δp =
~
.
2∆
(2.40)
A coherent state is the closest analogy to classical states one can achieve for a quantum harmonic
oscillator and is hence called quasi-classical state. The state evolution, a rotation in phase space (see
figure 2.2(a)), cannot be seen in ion trap experiments because the laser creating the displacement
acts as a reference oscillator rotating with the coherent state. How to create a displacement
operator with laser fields will be explained in chapter 3.
2.3. Atomic Structure
There are several ways to experimentally realize a good approximation of a two-level system.
Ideally the ion has to be completely separated from the environment and should only interact with
the laser. Experimentally this means, that the laser-ion coupling has to be much larger than any
coupling to the environment. A suitable choice for a qubit are either hyperfine/Zeeman ground
states or states connected by a forbidden transition. The element used to realize a qubit in this
thesis is the single charged alkali earth ion Ca+ . This section describes the relevant level structure
Ca+ as well as the qubits realized in these ions. A more detailed
description of 40 Ca+ can be found in [92] and [93] whereas for 43 Ca+ the reference is [94].
of the two isotopes
40
Ca+ and
43
13
Figure 2.3.: Detailed energy level scheme showing all Zeeman sublevels of the three lowest
orbitals of the 40 Ca+ ion. Laser light at 397 nm is used for Doppler-cooling, optical pumping
and detection. The lasers at 866 nm, 850 nm and 854 nm pump out the D-states. An
ultra-stable laser at 729 nm is used for manipulating the qubit encoded in the quadrupole
transition, state preparation, optical shelving and ground-state cooling. The qubit is encoded
in the states S1/2 (mj = 1/2) and D5/2 (mj = 3/2). The wavelength in air, natural lifetimes
τ and the branching fractions (denoted as XX% for the respective transition) are taken from
references [95–98].
Level scheme, Zeeman splitting and qubit choice
The coarse level structure for 40 Ca+ and 43 Ca+ is the same. The energy level schemes showing
the lowest three orbitals of the valence electron for 40 Ca+ and 43 Ca+ can be seen in figure 2.3
and figure 2.4. The S1/2 ground state is connected via a 397 nm dipole transition with the P1/2
state and a 393 nm dipole transition with the P3/2 state. The P-levels both having a lifetime of
about 7 ns exhibit a branching ratio such that one out of 14 decays populates the D-levels.1 To
avoid population trapping in the long lived D-states one can use the dipole transitions at 854 nm,
850 nm and 866 nm to pump out these levels. These wavelengths connect the two D-states to
either a P1/2 or P3/2 level via a dipole transition. A quadrupole transition at 729 nm connects the
S1/2 and the D5/2 level. Since this transition is dipole-forbidden, the D5/2 level is metastable and
has a lifetime of about 1.2 s which sets an upper time limit for the usefulness of a qubit encoded
in this state.
Calcium 40
In a magnetic field all levels split up into sublevels due to the Zeeman effect (see figure 2.3). The
splitting of the states is a few MHz for the magnetic fields typically used in experiments. Thus it
is well resolved for the quadrupole transition. For all the other transitions the splitting is hidden
within the linewidth of the transition. The S1/2 ground state splits up in two Zeeman sublevels with
mS = ±1/2 and the D5/2 level splits up in six Zeeman sublevels with mD = ±1/2, ±3/2, ±5/2. For
1 An
exact measurement of the P3/2 branching ratio measurements using QIP methods can be found in reference [95].
14
F
2
3
4
5
dhfs(MHz)
205.6(1.6)
117.5(0.8)
-4.5(0.8)
-164.5(1.1)
P3/2
3 327.2(2)
P1/2
4 -254.5(2)
854 nm
D5/2
866 nm
397 nm
D3/2
F
1
2
3
4
5
6
dhfs(MHz)
41.5267(6)
34.9516(4)
24.6348(2)
10.0312(3)
-9.5858(3)
-35.1246(4)
2
3
4
5
317.3(1.7)
178.0(0.8)
-10.1(0.9)
-249.3(1.2)
729 nm
3 1814.4046611(18)
S1/2
4 -1411.2036253(14)
Figure 2.4.: 43 Ca+ level scheme showing the hyperfine splitting of the lowest energy
levels. Hyperfine shifts δhf s of the levels are quoted in MHz (the splittings are taken from
[94, 99, 100]). Laser light at 397nm is used for Doppler-cooling, optical pumping and state
detection. The lasers at 866 nm, 850 nm and 854 nm pump out the D-states. An ultra stable
laser at 729 nm is used for manipulating the qubit encoded in the quadrupole transition,
state preparation, optical shelving and ground-state cooling.
a quadrupole transition changes in the magnetic quantum number of ∆m = 0, ±1, ±2 are allowed
by the selection rules. This results in a total of ten possible transitions from S1/2 to D5/2 .
One of these pairs of levels, the S1/2 , mj = 1/2 together with the D5/2 , mj = 3/2 was used as
a qubit for the experiments described in this thesis. The only exception to this choice are some
of the experiments presented in chapter 6.1 where the S1/2 , mj = 1/2 level together with the
D5/2 , mj = 5/2 level was used. These kinds of qubit are often termed optical qubits because the
transition frequency lies in the optical domain. As already mentioned, the sensitivity of the qubit
on the environment, here especially magnetic field fluctuations, is of importance. The dependence
of the transition frequency on the magnetic field B is given by
∆ωS→D = µB (ms gS1/2 − mD gD5/2 )B
(2.41)
where µB denotes the Bohr magneton and gS1/2 /gD5/2 the Lande-g-factor of the S1/2 /D5/2 level.
In the best case the lowest sensitivity one could get with respect to changes in the magnetic
field is 560kHz/G for the ∆m = 0 transitions which are usually used as a qubit. The employed
S1/2 , mj = 1/2 → D5/2 , mj = 3/2 transition has a factor of two higher dependence but comes with
the advantage that the coupling strength can be maximized by properly setting the polarization
and direction of the laser light with respect to the magnetic field (see chapter 3). This effectively
reduces the time needed for operations.
15
Calcium 43
The isotope 43 Ca+ is the only stable calcium isotope with a non-zero nuclear spin. It has a nuclear
spin of I = 7/2. Its level scheme is shown in figure 2.4. The coupling between electronic angular
momentum J and nuclear spin results in a hyperfine structure. The total angular momentum F
can take on the values |J − I| ≤ F ≤ |J + I|. For the ground state of 43 Ca+ we get two hyperfine
states with F=3 and F=4 split by a frequency difference of 3.225 GHz [99]. The D state with
J = 5/2 consists of 6 hyperfine states ranging from F=1 to F=6. In a magnetic field all hyperfine
levels split up in 2F + 1 Zeeman states.
This huge state manifold opens up several possibilities to encode a qubit. Two choices are
implemented in the experiments conducted in this thesis. An encoding in the two hyperfine ground
states termed hyperfine qubit or, similar to 40 Ca+ , an encoding using an optical qubit.
The first choice is interesting because there is no limitation by spontaneous decay. Furthermore
we can get a qubit which is insensitive to magnetic field fluctuations by choosing the mF = 0
sublevels of the two hyperfine ground states. This particular qubit was implemented in this thesis
(see chapter 6). From the Breit-Rabi formula [101] one can directly obtain that these sublevels
exhibit no first order Zeeman shift. However the linear dependence vanishes only for zero magnetic
field. For a finite field the differential shift is given by a second order approximation as [94]
∆ω0→0 =
(gJ − gI )2 µ2B 2
B ' 2π · 1.2 kHz/G2 · B 2
2~∆Ehf s
(2.42)
with the nuclear g-factor gI accounting for the complex structure of the nucleus and the Landé
factor
J(J + 1) + S(S + 1) − L(L + 1)
gJ ' 1 +
.
(2.43)
2J(J + 1)
This means that for a typical magnetic field of 4 G the magnetic field sensitivity is 19.2 kHz/G
which is much lower than for the best transition in 40 Ca+ . An additional advantage of hyperfine
qubits is that the transition frequency is in the GHz range so that the transition can be directly
addressed with a µ-wave field. Stable frequency sources with computer controlled power, phase
and frequency are readily available. If additional single ion addressing is required, one can switch
to a Raman laser setup as the microwave field cannot be focused. The two laser beams for the
Raman setup can be derived from the same laser source. The frequency splitting is achieved by
frequency shifting the light fields with acousto-optical modulators which ensures phase coherence.
An example for the implementation of a Raman laser with 43 Ca+ ions can be found in [94] and [102].
In this thesis the hyperfine qubit is solely manipulated by a microwave field.
The second choice, an optical qubit manipulated on the quadrupole transition is also appealing.
Although one is still limited by the finite lifetime of the D-state one can find transitions which
are insensitive to magnetic field fluctuations. This is due to the additional coupling of the nuclear
magnetic moment to the electron spin and the magnetic field which modifies equation (2.41). The
transition frequencies depend in a nonlinear way on the magnetic field which gives rise to a number
of different transitions which are insensitive to magnetic field fluctuations at a non zero magnetic
field. A good example is the S1/2 , F = 4, mF = 4 → D5/2 , F = 4, mF = 3 transition which
becomes field insensitive at 3.38 G and 4.96 G with a second-order Zeeman shift of ∓16kHz/G2 .
16
A much more detailed explanation of the hyperfine splitting in 43 Ca+ and the interaction with static
magnetic fields is given in [94]. Although the optical qubit in 43 Ca+ is insensitive to magnetic field
variations one should not forget that the coherence on all optical qubits is still limited by the
line width of the laser mediating the coupling. With the currently available quadrupole laser the
coherence time for a single ion is limited to about 8 ms (see chapter 4.6.2).
In the experiment two different qubits on the quadrupole transition were realized. One qubit
was realized using the levels S1/2 , F = 4, mF = 0 ↔ D5/2 (F = 6, mF = 1) at a magnetic field of
6 G. The second qubit was realized using the levels S1/2 , F = 4, mF = 0 ↔ D5/2 (F = 4, mF = 2)
at a magnetic field of 3.4 G. Both were chosen because they are separated from other transitions by
several MHz, have a moderate magnetic field dependence of 350 kHz/G and 220 kHz/G and high
coupling strength. Furthermore, their coupling strength can be maximized by properly setting the
polarization and direction of the laser light with respect to the magnetic field (see chapter 3).
17
18
3. Laser-ion interaction
This chapter summarizes the interactions between an ion trapped in a Paul trap and an incident
laser beam. The first two parts review resonant and non-resonant single-qubit interactions. The
following sections describe the interactions relevant for a quadrupole transition and a microwave
transition. Then the focus shifts to the implementation of a two-qubit operation by making use
of a bichromatic light field. This part was also published in [76]. Finally the bichromatic light
field will be used to create coherent displacement forces acting on the ion trap harmonic oscillator.
Furthermore, the bichromatic light field is used to measure the expectation values of position and
momentum for the harmonic oscillator.
3.1. Ion laser Hamiltonian
The Hamiltonian describing the trapped ion consists of two parts. The first part describes the ion
that can be treated as a two-level system with a laser tuned close to this transition. The second part
describes the ion in the external confining potential of the trap that forms a harmonic oscillator.
In the following description we assign a pseudo-spin to the qubit. The qubit states can then be
identified with |↓i (|↑i) for the lower(upper) energy level. For an ion with mass m oscillating with
frequency ω in the harmonic potential the system can be described by the following Hamiltonians
p2
1
+ mω 2 x2
2m 2
1
He = ~νσz
2
Hl = ~Ω(σ + + σ − )(ei(kx−νl t+φ) + e−i(kx−νl t+φ) )
Hm =
(3.1)
(3.2)
(3.3)
where σz , σ + = 12 (σx + iσy ), σ − = 12 (σx − iσy ) are Pauli spin operators and ν denotes the frequency
splitting of the qubit levels. Furthermore, Ω includes all details about the exact form of the atom
light interaction and is called the Rabi frequency, while k is the wave vector, νl the laser frequency
and φ the phase of the light field. Hm describes the motion of the ion in the harmonic potential,
while He describes the ions internal state. The laser-ion interaction is contained in the Hamiltonian
Hl . Using the creation and annihilation operators a† , a the Hamiltonians Hm and Hl can be
reexpressed as
1
1
H0 = Hm + He = ~ω(a† a + ) + ~νσz
2
2
†
†
Hl = ~Ω(σ + + σ − )(eiη(a+a ) e−i(νl t−φ) + e−iη(a+a ) ei(νl t−φ) )
19
(3.4)
(3.5)
Laser-ion interaction
Figure 3.1.: Combined energy levels of a two-level system and a harmonic oscillator. The
black arrows indicate carrier transitions whereas the red/blue arrows indicate the respective
sideband transitions.
Here the Lamb Dicke parameter η is used, which is defined as
r
η = ~k~ez
~
2mω
(3.6)
where ~ez denotes the oscillation axis. The Lamb Dicke parameter relates the spatial extension of
the harmonic oscillator ground state to the wavelength of the atomic transition. It describes the
ability of the light field to couple to the motion of the ion in the harmonic oscillator.
Transforming H = H0 + Hl into the interaction picture with Hint = U † HU , U = eiH0 t/~ and
using the rotating wave approximation leads to the Hamiltonian
Hint (t) = ~Ω(σ + eiη(ae
−iωt
+a† eiωt ) −i((νl −ν)t−φ)
e
+ h.c.).
(3.7)
Depending on the detuning ∆ = νl − ν of the laser from the transition frequency ν, it will couple
certain electronic and motional states. If the detuning is exactly ∆ = s ω it will couple the state
|↓, ni to |↑, n + si, where n is the vibrational quantum number and s ∈ N0 . Equation (3.7) contains
all details about the laser ion interaction.
3.1.1. Lamb Dicke regime
An ion is said to be in the Lamb Dicke regime if the extension of its wavefunction is small compared
to the wavelength of the light. In this regime, the inequality η 2 xrms ¿ 1 is fulfilled with xrms the
extent of the vibrational mode’s wave function. Then equation (3.7) can be simplified by a Taylor
expansion of
−iωt
+a† eiωt )
eiη(ae
= 1 + iη(ae−iωt + a† eiωt ) + O(η 2 ).
(3.8)
Only first order terms need to be considered, that is transitions which change the vibrational
quantum number by no more than ±1. The interaction Hamiltonian is now given by
Hint (t) = ~Ωσ + (1 + iη(ae−iωt + a† eiωt ))e−i((νl −ν)t−φ) + h.c..
(3.9)
This Hamiltonian gives rise to three different resonances:
• For νl = ν only the electronic state of the ion is altered while ∆n = 0. The transition
(|↓, ni → |↑, ni is called carrier transition where the coupling strength, including a second
20
3.2 Non-resonant interactions
order term, scales as Ωn,n = (1 − η 2 n)Ω. The small correction ≈ n is obtained going beyond
the approximation (3.9). The effective Hamiltonian is given by
Hcar (t) = ~Ωn,n (σ + eiφ + σ − e−iφ ).
(3.10)
Depending on the angle φ one can implement σx , σy interactions realizing two of the three
single qubit operations described in chapter 2.1.2.
• For νl = ν − ωz the electronic state of the ion is changed and n decreases by one. The laser
couples the states |↓, ni → |↑, n − 1i. The coupling strength for this red sideband transition
√
scales as Ωn−1,n = η nΩ. The Hamiltonian is given by
Hred (t) = i~Ωn−1,n (aσ + eiφ − a† σ − e−iφ ).
(3.11)
• Similar for νl = ν + ωz the electronic state of the ion is changed and n increases by one.
The laser couples |↓, ni → |↑, n + 1i. The coupling strength for this blue sideband transition
√
scales as Ωn+1,n = η n + 1Ω. The Hamiltonian is given by
Hblue (t) = i~Ωn+1,n (a† σ + eiφ − aσ − e−iφ ).
(3.12)
A graphical illustration of carrier and sideband transitions is shown in figure 3.1. It shows the
combined energy levels of a two-level system and a harmonic oscillator.
With the help of sideband transitions one can convert the internal excitation of one ion into a
collective motion of the whole ion string. For example the state |↑↓ . . . ↓, 0i, carrying one electronic
excitation, will be transformed into the state |↓↓ . . . ↓, 1i, carrying one phonon, by a π-pulse on
the red sideband. This excitation can be transferred onto any other ion in the string by another
red sideband pulse. In this way two or more ions can interact with another. The motion acts as a
kind of bus system mediating the interaction.
3.2. Non-resonant interactions
So far we have only considered resonant interactions. However, also non-resonant interactions
play a crucial role in the ion-light interaction. On one hand, they can be used to create desired
interactions, on the other hand they cause unwanted effects. Non-resonant interactions become
especially important when driving sideband transitions. Due to the small coupling of the sidebands
to the laser a lot of light is needed. This causes a frequency shift on the atomic levels due to the
carrier and population transfer on the carrier.
3.2.1. Light shifts
A light field which is off-resonant from a transition will shift the atomic levels. This effect is known
as AC-Stark-effect. Consider a two-level atom with an unperturbed energy difference E = ~ν and
a detuning of the light field by δ = ν − νl from this transition. The amplitude of the light field
is given by a Rabi frequency Ω. The level shift is calculated by going along the same line of
21
Figure 3.2.: (a) Rabi oscillations on the sideband (red) with a Rabi frequency Ω =
2π · 45 kHz. The black line with an amplitude of 0.025 are Rabi oscillations with a frequency
of 1.2 MHz taking place on the carrier transition. The oscillation frequency and amplitude
is given by equation (3.15). (b) Pulse form of a shaped pulse. The slopes are shaped as a
Blackman window, the figure shows the definitions of pulse and slope duration.
reasoning as done for equation (3.7) and calculating the new eigenenergies of the system from the
Hamiltonian
~δ
Hstark =
σz + ~Ωσx .
(3.13)
2
Here only carrier transitions were taken into account. For δ >> Ω one finds that the upper level
will be shifted by Ω2 /δ and the lower level will be shifted by −Ω2 /δ. This amounts to a total
frequency shift of the transition by
Ω2
δstark = 2 .
(3.14)
δ
From the Hamiltonian one can already see that such an interaction is proportional to a σz rotation
on a qubit. Thus by shining in a detuned light field for a certain amount of time one can perform
an arbitrary rotation around the z-axes realizing the third single qubit operation mentioned in
chapter 2.1.2.
So far it seems that Stark shifts are only desired interactions. This changes when one considers exciting sideband transitions like |↓, 0i → |↑, 1i which are needed for some two-qubit gate
implementations [14]. These gate-operations are similar to interference experiments, thus it is of
utmost importance that the phase relation between pulses does not vary over time. Due to the
AC-Stark shift generated by the carrier, intensity fluctuations would be immediately transferred
into phase fluctuations and thus spoil the quality of the gate operation. A possibility to get rid of
these amplitude-dependent Stark shifts is to shine in a second light field detuned in opposite direction with a Rabi frequency exactly canceling the effect. This light field is derived from the same
beam as the driving field and thus has the same amplitude fluctuations which together leads to a
common-mode rejection of the Stark shift. A thorough explanation and analysis of this technique
can be found in [93, 103].
3.2.2. Off resonant excitation
A light field resonant with the transition |↓, 0i → |↑, 1i will not only drive Rabi oscillations on
this transition but also off-resonantly drive the carrier transition |↓, 0i → |↑, 0i. This effect can be
understood by looking at the time evolution of the |D, 0i population, which is obtained by solving
22
3.3 The S1/2 → D5/2 transition
the optical Bloch equations [104]
µ q
¶
1
2
2
=
sin
(2Ω) + ∆ t .
2
2
(2Ω) + ∆2
2
P|↑,0i
(2Ω)
2
(3.15)
Typical experimental parameters are Ω = 2π · 45 kHz and an axial trap frequency and thus
detuning of ∆ = ωax = 2π · 1.2 MHz. This will result in off-resonant oscillations with an amplitude
of 2.5% and a frequency of 1.2 MHz. Rabi oscillation on the carrier and the sideband are shown in
figure 3.2(a). As gate operations have to reach fidelities exceeding 99% one has to find a method
to avoid this effect.
The method of choice is to switch pulses adiabatically as shown in figure 3.2(b). This pulse
shaping has two effects. First, it suppresses high frequency components in the Fourier transform of
the pulse which are related to the off-resonant excitations. Second, due to the adiabatic switching
which starts and stops with a zero amplitude of the pulse, the off-resonant excitations are adiabatically transformed to zero. Typical pulse slopes are 5 to 10 µs for a light power equivalent to
performing a full rabi oscillation within 100 µs on the blue sideband. Further information on pulse
shaping and off-resonant excitations can be found in [105].
3.3. The S1/2 → D5/2 transition
For all but one experiment in this thesis the dipole forbidden transition S1/2 → D5/2 of 40 Ca+
and 43 Ca+ was used to realize a qubit. In this section the excitation of the quadrupole transition
S1/2 ↔ D5/2 by laser light is discussed which is important to understand the experiments presented
in chapters 6, 7 and 8.
3.3.1. Rabi frequency and geometrical considerations
For a quadrupole transition, the induced electric quadrupole moment Q couples to the gradient of
the applied electric field E. The Hamiltonian describing this interaction is given by
HI = −Q · ∇E(t).
(3.16)
This type of interaction takes on the shape of equation (3.3) when the Rabi frequency is defined
as
¯
E¯¯
¯ eE0 D
0
0 ¯
~
¯
S1/2 , F, mF |(~² · ~r)(k · ~r)|D5/2 , F , mF ¯
Ω=¯
(3.17)
4~
with E0 the electric field amplitude and ~r the position operator of the valence electron relative
to the atomic nucleus. For 40 Ca+ the quantum numbers F, F 0 can be ignored and mF , m0F be
replaced by mj , m0j . The selection rules allow for ∆m = mj − m0j = 0, ±1, ±2. In the case of
43
Ca+ , the same rules apply for mF , m0F but additionally ∆F = F − F 0 = 0, ±1, ±2 has to be
fulfilled. From the term (~² · ~r)(~k · ~r) one can already guess that the effective coupling strength
depends on the laser beam direction and polarization relative to the quantization axis defined by
23
the magnetic field vector. The expression for the Rabi frequency can be rewritten in the form [94]
eE0
Ω=
4~
r
15 ΓD5/2
Λ(F, m, F 0 , m0 ) g ∆m (φ, γ)
cα k 3
(3.18)
with the fine structure constant α, the speed of light c, the electron charge e and the spontaneous
decay rate ΓD5/2 . The coupling strength Λ(F, m, F 0 , m0 ) contains all properties of the atomic
transition apart from geometrical factors. The coefficients for all transitions in 40 Ca+ and 43 Ca+
are listed in Appendix C.
The function g ∆m (φ, γ) contains the geometry dependence of Ω [92, 98] where φ denotes the
angle between the laser beam and the magnetic field and γ describes the angle between a linear
~ =
polarization and the magnetic field vector projected onto the plain of incidence. Choosing B
1
~
B0 (0, 0, 1), one obtains for k = k(sin φ, 0, cos φ) and ~² = (cos γ cos φ, sin γ, − cos γ sin φ)
1
| cos γ sin(2φ)|
2
1
= √ | cos γ cos(2φ) + i sin γ cos φ|
6
1 1
= √ | cos γ sin(2φ) + i sin γ sin φ|
6 2
g (0) =
g (±1)
g (±2)
These terms change for circularly polarized light with ~² =
for σ ± polarized light to
√1 (p
2
(3.19)
(3.20)
(3.21)
· i cos φ, 1, −p · i sin φ) and p = ±1
1
g (0) = √ | sin(2φ)|
8
1
g (±1) = √ | cos(φ) − ∆m · p · cos(2φ)|
12
1
1
g (±2) = √ | sin(2φ) + sin φ|
12 2
(3.22)
(3.23)
(3.24)
Since any Zeeman transition from S1/2 → D5/2 can be used for coherent manipulation of the ions’
quantum state, it is interesting to look at the possibility to change the coupling strengths of these
transitions. Three cases are especially useful when it comes to maximizing the coupling strength
of a particular transition and minimize others:
• Linear polarized light with φ = 90◦ , γ = 90◦ . Exciting only ∆m = +2 and ∆m = −2
transitions. This configuration could be used for sideband cooling (see chapter 5).
• Linear polarized light with φ = 45◦ , γ = 0◦ . Here the ∆m = 0 transitions couple
strongest to the laser while ∆m = ±1 are completely suppressed. In this configuration one
would use e.g. the 40 Ca+ S1/2 (m = +1/2) → D5/2 (m = +1/2) transition for storing the
qubit while doing the sideband cooling on S1/2 (m = +1/2) → D5/2 (m = +5/2).
• Circular polarized light along the magnetic field axis φ = 0◦ . This is the only
configuration where a single transition can be selected by using σ± polarized light to excite
∆m = ±1 transitions. This also results in the biggest coupling strength that can be achieved.
1 the
y-component can be chosen to be zero due to cylindrical symmetry
24
3.4 Microwave transitions
A disadvantage of this configuration is that sideband cooling is not possible as the cooling
cycle can only be closed via the P3/2 (mj = +3/2) state when the S1/2 (mj = +1/2) →
D5/2 (mj = +5/2) transition is excited.
Configuration 3 is especially interesting in
43
Ca+ to reduce the number of possible transitions by
a considerable amount. Thereby one can also suppress unwanted off-resonant excitation and Stark
shifts due to other carrier transitions close by. A more mathematical treatment of this subject
with equations that are valid for arbitrary polarization and angle can be found in [92].
3.4. Microwave transitions
For the hyperfine ground state manifold of 43 Ca+ microwave transitions are of importance to
manipulate the qubit. A microwave field couples to the magnetic dipole moment of the ion. The
Rabi frequency can be expressed as
¯
¯
¯
¯1
¯
®¯
~ M W ¯S1/2 , F 0 = 3, m0F ¯
ΩM W = ¯¯
S1/2 , F = 4, mF ¯ µ
~BB
¯
2~
(3.25)
~ M W the magnetic field of the microwave radiation.
with µ
~ B the ion’s magnetic dipole moment and B
Similar to dipole transitions one can drive a ∆mF = 0 transition with a π-polarized AC magnetic
field, whereas for ∆mF = ±1 transitions σ± -polarized fields are needed.
With microwave fields it is impossible to directly drive sideband transitions as the Lamb-Dicke
parameter is extremely small. For trapping frequencies of 1 MHz and the ground state hyperfine
splitting in 43 Ca+ of 3.2 GHz the Lamb-Dicke parameter is on the order of 10−6 . Another drawback
is that microwave fields can not be used to address single ions. The wavelength is so big that no
focussing can be achieved with respect to the few µm distance of the ions. Both problems might
be circumvented by either using high magnetic field gradients [106] and/or oscillating magnetic
fields [107]. These approaches are quite challenging and were not pursued in this thesis.
Although microwave transitions have drawbacks they are a valuable tool for reliably driving the
hyperfine qubit, especially if all ions in the trap are subject to the same operation. In this thesis
the microwave field was used to map between the hyperfine and the optical qubit (see chapter 6)
and to characterize the coherence properties of the hyperfine qubit.
3.5. Mølmer-Sørensen gate
A two-qubit quantum gate-operation that is equivalent to a controlled-NOT gate up to local
operations is achieved by the action of a Hamiltonian H ∝ σn ⊗σn , where σn = σ · n is a projection
of the vector of Pauli spin matrices onto the direction n [108]. Two prominent examples of this type
of gate are the conditional phase gate [109, 110] and the Mølmer-Sørensen gate [34, 80, 111] (MSgate). In the latter case, correlated spin flips between the states |↑i |↑i ↔ |↓i |↓i and |↑i |↓i ↔ |↓i |↑i
are induced by a Hamiltonian
H ∝ σφ ⊗σφ
where
σφ = cos φ σx + sin φ σy .
25
(3.26)
Figure 3.3.: Mølmer-Sørensen interaction scheme. A bichromatic laser field couples the
qubit states |↓↓i ↔ |↑↑i via the four interfering paths shown in the figure. Similar processes
couple the states |↑↓i ↔ |↓↑i. The frequencies ν± of the laser field are tuned close to the
red and the blue motional sidebands of the qubit transition with frequency ν0 , and satisfy
the resonance condition 2ν0 = ν+ + ν− . The vibrational quantum number is denoted n.
The unitary operation UφM S = exp(i π4 σφ ⊗ σφ ) maps product states onto maximally entangled
states. In 1999 the proposal was made to realize an effective Hamiltonian [80, 111] taking the
form (3.26) by exciting both ions simultaneously with a bichromatic laser beam with frequencies
ν± = ν0 ±δ where ν0 is the qubit transition frequency and δ is close to a vibrational mode of the twoion crystal with frequency ω. Figure 3.3 shows the level scheme of two ions in a harmonic oscillator
and the laser fields applied for a Mølmer-Sørensen interaction. Changing into an interaction picture
and performing a rotating-wave approximation, the time-dependent Hamiltonian
H(t) = ~Ω(e−iδt + eiδt )eiη(ae
−iωt
+a† eiωt )
(1)
(2)
(σ+ + σ+ ) + h.c.
(3.27)
is well approximated by
H(t) = −~ηΩ(a† ei(ω−δ)t + ae−i(ω−δ)t )Sy
(3.28)
in the Lamb-Dicke regime.
(2)
(1)
In equation (3.28), we use a collective spin operator Sy = σy + σy . In the following we denote
the laser detuning from the motional sidebands by ω − δ = ². The whole subsequent treatment
PN (i)
of the MS-interaction can be generalized for N ions by exchanging Sy with SyN = i=0 σy . The
Hamiltonian (3.28) can be exactly integrated [81] yielding the propagator
U (t) = D̂(α(t)Sy )e(i(λt−χ sin(²t))Sy ) ,
2
(3.29)
†
∗
i²t
− 1), λ = η 2 Ω2 /², χ = η 2 Ω2 /²2 , and D̂(α) = eαa −α a is a displacement
where α(t) = ηΩ
² (e
operator. For a gate time tgate = 2π/|²|, the displacement operator vanishes so that the propagator
U (tgate ) = exp(iλtgate Sy2 ) can be regarded as being the action of an effective Hamiltonian
Heff = −~λSy2 = −2~λ(1 + σy ⊗ σy )
(3.30)
inducing the same action up to a global phase as the one given in (3.26). Setting Ω = |²|/(4η), a
gate is realized, which is capable of maximally entangling ions irrespective of their motional state.
In the description of the gate mechanism given so far, a coupling of the light field to the carrier
transition was neglected based on the assumption that the Rabi frequency Ω was small compared
26
3.5 Mølmer-Sørensen gate
(a)
(b)
z=0
z = p/2
gate operation
gate operation
z
z
y
y
x
x
Figure 3.4.: Effect of non-resonant excitation of the carrier transition. (a) For ζ = 0,
the gate starts at a maximum of the intensity-modulated beam. In this case, a Bloch
vector initially centered at the south pole of the Bloch sphere performs oscillations that are
symmetric around the initial position. (b) For ζ = π/2, the gate starts at the minimum of
the intensity modulation. In this case, the average orientation of the Bloch vector is tilted
with respect to its initial position.
to the detuning δ of the laser frequency components from the transition. In this case, small nonresonant Rabi oscillations that appear on top of the gate dynamics are the main effect of coupling to
the carrier transition. Since a maximally entangling gate requires a Rabi frequency Ω ∝ η −1 t−1
gate ,
the question of whether Ω ¿ δ holds becomes crucial in the limit of fast gate operations and
small Lamb-Dicke factors. Our experiments [40] are exactly operating in this regime, and it turns
out that non-resonant excitation of the carrier transition has further effects beyond inducing nonresonant oscillations [112]. This becomes apparent by interpreting terms in the Hamiltonian in a
different way: The red- and blue-detuned frequency components E± = E0 cos((ν0 ±δ)t±ζ) of equal
intensity may be viewed as a single laser beam E(t) = E+ + E− = 2E0 cos(ν0 t) cos(δt + ζ) that is
resonant with the qubit transition but amplitude-modulated with frequency δ. Here, the phase ζ
which determines whether the gate operation starts in a maximum (ζ = 0) or a minimum (ζ = π/2)
of the intensity of the amplitude-modulated beam has a crucial influence on the gate-operation.
This can be intuitively understood by considering the initial action the gate exerts on an input
state in the Bloch sphere picture shown in figure 3.4. For short times, coupling to the sidebands
can be neglected which justifies the use of a single-ion picture. The dynamics is essentially the
one of two uncoupled qubits. The fast dynamics of the gate is induced by excitation of the ions
on the carrier transition. For ζ = 0, the Bloch vector of an ion initially in state |↓i will oscillate
with frequency δ along a line centered on the south pole of the Bloch sphere. For ζ = π/2, the
oscillation frequency is the same, however, the time-averaged position of the Bloch vector is tilted
by an angle
4Ω
ψ=
sin ζ
(3.31)
δ
with respect to the initial state |↓i. This effect has a profound influence on the gate action.
A careful analysis of the gate mechanism [112] taking into account the non-resonant oscillations
reveals that the Hamiltonian (3.28) is changed into
H(t) = −~ηΩ(a† ei(ω−δ)t + ae−i(ω−δ)t )Sy,ψ ,
27
(3.32)
where
Sy,ψ = Sy cos ψ + Sz sin ψ ,
(3.33)
and that the propagator (3.29) needs to be replaced by
¡
¢
2
U (t) = e(−iF (t)Sx ) D̂(α(t)Sy,ψ )e i(λt − χ sin(²t))Sy,ψ
,
(3.34)
where the term containing F (t) = 2Ω
δ (sin(δt + ζ) − sin ζ) describes non-resonant excitation of
the carrier transition. In the derivation of Hamiltonian (3.32), small terms arising from the noncommutativity of the operators Sy , Sz have been neglected [112]. The dependence of the propagator
on the exact value of ζ is inconvenient from an experimental point of view. To realize the desired
gate-operation in an optimal way, precise control over ζ is required. In addition, the gate duration
must be controlled to better than a fraction of the mode oscillation period because of the nonresonant oscillation. Fortunately, both of these problems can be overcome by shaping the overall
intensity of the laser pulse such that the Rabi frequency Ω(t) smoothly rises within a few cycles 2π/δ
to its maximum value Ωgate ≈ |²|/(4η) and smoothly falls off to zero at the end of the gate. In this
case, the non-resonant oscillations vanish and (3.31) shows that the operator Sy,ψ (t) adiabatically
follows the laser intensity so that it starts and ends as the desired operator Sy irrespective of
the phase ζ. For intensity-shaped pulses, the propagator (3.29) provides therefore an adequate
description of the gate action.
3.5.1. AC-Stark shifts
In the description of the gate mechanism given so far the ion was treated as an ideal two-level
system. AC-Stark shifts are completely insignificant provided that the intensities of the blue- and
the red-detuned frequency components are the same since in this case light shifts of the carrier
transition caused by the blue-detuned part are exactly canceled by light shifts of the red-detuned
light field. Similarly, light shifts of the blue-detuned frequency component non-resonantly exciting
the upper motional sideband are perfectly canceled by light shifts of the red-detuned frequency
component coupling to the lower motional sideband.
For an experimental implementation with calcium ions, we need to consider numerous energy
levels (see figure 2.3). Here, the laser field inducing the gate action causes AC-Stark shifts on the
qubit transition frequency due to non-resonant excitation of far-detuned dipole transitions and also
of other S1/2 ↔ D5/2 Zeeman transitions. The main contributions arise from couplings between
the qubit states and the 4p−states that are mediated by the dipole transitions S1/2 ↔ P1/2 ,
S1/2 ↔ P3/2 , D5/2 ↔ P3/2 . Other transitions hardly matter as can be checked by comparing the
experimental results obtained in [103] with numerical results based on the transition strengths [98]
of the dipole transitions coupling to the 4p−states. For suitably chosen k-vector and polarization
of the bichromatic laser beam, these shifts are considerably smaller than the strength λ of the gate
interaction.
AC-Stark shifts can be compensated by a suitable detuning of the gate laser. An alternative
strategy consists in introducing an additional AC-Stark shift of opposite sign that is also caused by
the gate laser beam [103]. This approach has the advantage of making the AC-Stark compensation
independent of the gate laser intensity. In contrast to previous gate-operations relying on this
28
3.5 Mølmer-Sørensen gate
technique [113] where the AC-Stark shift was caused by the quadrupole transition and compensated
by coupling to dipole transitions, here, the AC-Stark shift is due to dipole transitions and needs
to be compensated by coupling to the quadrupole transition.
For ions prepared in the ground state of motion (n = 0), a convenient way of accomplishing this
task is to perform the gate operation with slightly imbalanced intensities of the blue- and the reddetuned laser frequency components. Setting the Rabi frequency of the blue-detuned component
to Ωb = Ω(1 + ξ) and the one of the red-detuned to Ωr = Ω(1 − ξ), an additional light shift caused
(C)
by coupling to the carrier transition is induced that amounts to δac = 2(Ω2r − Ω2b )/δ = −8Ω2 ξ/δ.
Now, the beam imbalance parameter ξ needs to be set such that the additional light shift exactly
cancels the phase shift φ = δac tgate induced by the dipole transitions during the action of the gate.
Taking into account that tgate = 2π/² and Ω = |²|/(4η), this requires ξ = (δη 2 /|²|)(φ/π).
Apart from introducing light shifts, setting ξ 6= 0 also slightly changes the gate Hamiltonian
(3.30) from Heff = −λSy2 to Heff = −λ(Sy2 + ξ 2 Sx2 ) [114] since the coupling between the states |↓↓i,
|↑↑i is proportional to 2Ωb Ωr = 2Ω2 (1 − ξ 2 ) whereas the coupling between |↓↑i, |↑↓i is proportional
to Ω2b + Ω2r = 2Ω2 (1 + ξ 2 ). However, as long as ξ ¿ 1 holds – which is the case in the experiments
described in chapter 6 – this effect is extremely small2 as the additional term is only quadratic in
ξ.
Another side effect of setting ξ 6= 0 is the occurrence of an additional term ∝ Sz a† a in the
Hamiltonian. It is caused by AC-Stark shifts arising from coupling to the upper and lower motional
sideband which no longer completely cancel each other. The resulting shift of the qubit transition
frequency depends on the vibrational quantum number n and is given by δ (SB) = (8η 2 Ω2 /²)ξn =
(²/2)ξn. Simulations of the gate action based on (3.27) including an additional term ∝ Sz accounting for AC-Stark shifts of the dipole transitions and power-imbalanced beams show that the
unwanted term ∝ Sz a† a has no severe effects for ions prepared in the motional ground state as long
as ξ ¿ 1. However, for ions in Fock states with n > 0, this is not the case. Taking the parameter
set ξ = 0.075, ω = (2π) 1230 kHz, and ² = (2π) 20 kHz as an example, the following numerical
results are obtained: applying the gate to ions prepared in | ↓↓i|n = 0i, a Bell state is created
with fidelity 0.9993. For n = 1, the fidelity drops to 0.985, and for n = 2 to even 0.942. This loss
of fidelity can be only partially recovered by shifting the laser frequency by δ (SB) , the resulting
fidelity being 0.993 and 0.968, for n = 1 and n = 2 respectively. For higher motional states, the
effect is even more severe and shows that this kind of AC-Stark compensation is inappropriate
when dealing with ions in a thermal state of motion with n̄ À 1. Instead of compensating the
AC-Stark shift by imbalancing the beam powers, in this case, the laser frequency needs to be
adjusted accordingly (see chapter 6 on the experiments with Doppler-cooled ions).
3.5.2. A Mølmer-Sørensen interaction with ions in a thermal state of motion
In theory, the Mølmer-Sørensen gate does not require the ions to be cooled to the ground states
of motion since its propagator (3.29) is independent of the vibrational state for t = tgate . For
additional term Sx2 changes the gate operation from U = exp (−i π8 Sy2 ) to Uξ = exp (−i π8 (Sy2 + ξ 2 Sx2 )). A
short calculation shows that the minimum state fidelity Fmin = min{ψ} (|hψ|U † Uξ |ψi|2 ) is given by Fmin =
1
2 . Thus, for ξ = 0.075, one
(1 + cos( π2 ξ 2 )) where we used [Sx2 , Sy2 ] = 0 and exp(−iγSx2 ) = 1 + 14 (e−i4γ − 1)SX
2
−5
obtains Fmin ≈ 1 − 2 · 10 .
2 The
29
t 6= tgate , however, the interaction entangles qubit states and vibrational states so that the qubits’
final state becomes dependent on the initial vibrational state. Therefore, it is of interest to calculate
expectation values of observables acting on the qubit state space after applying the propagator for
an arbitrary time t. Simple expressions can be derived [76] in the case of a thermally occupied
motional state where the mode population probabilities are given by
p̃n =
1
n̄ + 1
µ
n̄
n̄ + 1
¶n
.
(3.35)
Not taking into account non-resonant carrier oscillations, one finds the following expressions for
the qubit state populations
2
2
1
1
1
(3 + e−16|α| (n̄+ 2 ) + 4 cos(4γ)e−4|α| (n̄+ 2 ) )
8
2
1
1
p↑↓ (t) + p↓↑ (t) = (1 − e−16|α| (n̄+ 2 ) )
4
2
2
1
1
1
p↓↓ (t) = (3 + e−16|α| (n̄+ 2 ) − 4 cos(4γ)e−4|α| (n̄+ 2 ) )
8
p↑↑ (t) =
(3.36)
with α = α(t) and γ = λt − χ sin(²t) containing the time dependent terms.
3.6. Creating a coherent displacement
Another application for bichromatic light fields is to create coherent displacement forces. This can
be done by shining in two light fields resonant with the blue and red sideband transition. Putting
together equations (3.12) and (3.11) and calculating a few lines to simplify the expressions one
arrives at the following Hamiltonian
HD = ~ηΩ [σx cos φ+ − σy sin φ+ ]
£
¤
⊗ (a + a† ) cos φ− + i(a† − a) sin φ− .
(3.37)
Here 2φ+ = φr + φb and 2φ− = φb − φr are the sum and the difference, respectively, of the phases
of the light fields tuned to the red and blue sideband. One can recognize the position x̂ and
momentum operators p̂ in the second part of the Hamiltonian. The propagator for φ− = φ+ = 0
is then
−iηΩt
UD = e ∆ xσx
(3.38)
which describes a state dependent coherent displacement operator D(ασx ) with α = iηΩt/∆ (see
chapter 2). The propagator (3.38) displaces an eigenstate of σx along p in phase space. If the
initial state was in the ground state a coherent state is created. By properly choosing the phases
φ− , φ+ one can create coherent states with arbitrary α.
3.6.1. Measurement of hxi and hpi
A lot of ion trap experiments make extensive use of the harmonic oscillator created by the trap
confinement for implementing two-qubit interactions. The ion trap system is always referred to as
30
3.6 Creating a coherent displacement
“the“ toy model for a quantum harmonic oscillator. Nevertheless the expectation values for the
position and momentum operators hxi and hpi have never been measured with trapped ions up to
now. Here a method is presented similar to one proposed in [115–117] to measure these quantities
without reconstructing the full quantum state.
As already mentioned in chapter 2.1.3 the only observable that can be directly measured in ion
trap experiments by fluorescence detection is σz . By using additional laser pulses one can generate
a propagator U that maps other observables onto σz . In this case we apply the propagator UD to
the state ρ of the ion prior to the state detection which is equivalent to measuring the observable
†
A(t) = UD
σz UD = cos(2ηx̂Ωp t/∆)σz + sin(2ηx̂Ωp t/∆)σy
(3.39)
on the initial state because of Tr((U † ρU )σz ) = Tr(ρ(U σz U † )). If the ion’s internal initial state is
the eigenstate of σz belonging to eigenvalue +1, hA(t)i = cos(2ηx̂Ωp t/∆) and for the eigenstate of
σy belonging to eigenvalue +1, hA(t)i = sin(2ηx̂Ωp t/∆).
Thus preparing ρ in an eigenstate of σy and taking the first derivative of the measured expectation
value
¯
¯
d
hA(t)i¯¯
= 2ηΩp /∆hx̂σy i
(3.40)
dt
t=0
reveals that the slope of the measured excitation at t = 0 is proportional to hxi. By changing the
phase φ− from 0 to π/2 it is possible to measure hpi instead of hxi. Furthermore all n-momenta of
x can be measured by taking the n-th derivative
¯
¯
dn
¯
hA(t)i
∝ hx̂n σy i.
¯
dtn
t=0
(3.41)
We have applied this technique to determine position and momentum in a simulation of a Dirac
particle encoded in position and momentum of the harmonic oscillator (see chapter 8).
31
32
4. Experimental setup
In this chapter an experimental setup is described used to realize the experiments discussed in the
proceeding chapters. A section about the ion trap and the vacuum setup will be followed by a
description of the laser system and the computer control.
4.1. Linear ion trap and electrode wiring
The ion trap used in this thesis is the standard Innsbruck design (see figure 4.1(a)). It is a
linear Paul trap where the radio frequency (RF) drive is applied to one of two pairs of blades
while the other pair is grounded. This creates the radial confinement. An additional DC voltage
applied to the tips generates the axial confinement (see figure 4.1(b)). A thorough discussion of
the specifications and design of the trap can be found in Jan Benhelm’s thesis [94].
The ion trap is operated at a frequency of about 25.5 MHz produced by a signal generator1 whose
output is amplified2 to 5-13 W. This signal is then coupled into a helical resonator amplifying the
voltage to about 1200 V which is applied to the blades. This creates a harmonic potential with
trapping frequencies between 2.5 MHz and 4 MHz in the radial directions. The two radial modes
are almost degenerate in frequency and measurements showed that they are split by about 50 kHz.
The orientation of these modes is along the RF-blades of the trap. This breaking of the radial
symmetry is due to the fact that the RF potential is only applied to one blade pair while the other
is grounded. The DC voltage applied to the tips ranges between 800-1400 V. A typical voltage of
1200 V creates an axial potential with a trap frequency of 1.36 MHz for a 40 Ca+ ion.
1 Rohde
2 Mini
& Schwarz, SML01
Circuits LZY-1
Figure 4.1.: (a) Picture of the linear Paul trap used in the experiment. (b) Schematic
drawing of the trap. Two opposite blade shaped electrodes (A) are connected to a high
voltage radio frequency while the other pair is grounded. The tips (B) are connected to a
DC voltage up to 1400 V. The compensation electrodes (C) are used to shift the ions into
the RF null compensating stray electric fields. The upper compensation electrode of the
pair used for horizontal micromotion compensation is also used to guide microwave signals
to the ions. The distance of the tips and electrodes is given in mm.
33
Experimental setup
Figure 4.2.: Wiring of the DC-trap electrodes. Each tip is wired to a circuit that can
change the voltages on the tips by twice the preset voltages U1/U2. A TTL signal switches
between the two voltages and the RC combination is designed such that voltage reaches the
desired value within 40 µs. This time constant is given by the low pass filters attached to
the tips. The virtual ground Ug for the secondary side is provided by a stable high voltage
source (ISEG). The effective voltage on the tips is then given by Utip1 = Ug ± U1/2 and
Utip2 = Ug ∓ U1/2 .
These voltage values lead to a string of ions in the trap along the direction of the tips. The
equilibrium positions of the ions are determined by the external trapping potential and the Coulomb
repulsion between the ions. For two ions of mass m the distance can be calculated analytically and
is given by
µ
¶1/3
e2
∆z =
.
(4.1)
2π²0 mωz
For an axial center of mass frequency ωz =1.2 MHz this leads to an ion spacing of 4.6 µm. This
knowledge is used to calibrate the imaging system in combination with the precisely measurable
trap frequency.
Each tip is wired separately such that it is possible to put them on different potentials. By
changing the potentials applied to the tips the ion string can be moved along the trap axis. To
shuttle the ions fast without compromising the stability of the trap potential we designed a circuit
which can change the tip voltage by ±20 V. With a TTL signal one can switch between two preset
values U1 , U2 such that the voltage on the tips changes within 40 µs. The time constant is given
by the low pass filters attached to the tips. On the one hand these filters block noise going to
the trap electrodes and on the other hand they shield the electronics from RF pick up by the
tips. A schematic drawing of the circuit can be seen in figure 4.2. A stable high voltage3 source
provides the virtual ground (Ug = 800 − 1200 V ) for the secondary side of the analog optocoupler4 .
A power supply referenced to the same high voltage provides the necessary current to drive the
adder/multplier circuit. This design allows one to combine the stability of the high voltage box
< 10−5 with a fast switching capability. The instability of the voltages U1 , U2 (∆V /V ≈ 10−3 )
and the noise of the electronic circuit (∆V /V ≈ 10−3 ) do not harm. This comes from the fact,
that the noise has to be compared to the 800 to 1200 V provided by the high voltage source. The
effective noise is less than 10−5 which is sufficient for all experiments in this thesis. The effective
voltage on the tips is given by Utip1 = Ug ± U1/2 and Utip2 = Ug ∓ U1/2 .
Due to electric stray fields, the ion position can be pushed out of the RF null leading to a motion
of the ion with a frequency which is the same as the drive frequency. This excess micromotion
3 ISEG,
EHQ F020p, 0-2kV, ∆V /V < 10−5
Agilent Tech.
4 HCNR201
34
4.2 Vacuum vessel
Figure 4.3.: (a) Schematic drawing of the trap setup viewed from atop showing trap
orientation, laser beam directions and positioning of the inverted viewports. Most of the
laser beams entering the chamber are within the equatorial plane. Two coils in almost
Helmholtz configuration generate a magnetic field defining the quantization axes along SWNE. An additional coil pair along SE-NW is used for compensating external magnetic stray
fields. The inverted view ports allow us to place custom-made lenses close to the ions. The
lenses are used to focus laser beams tightly and at the same time collect a big fraction of
the fluorescence light emitted by the ions. (b) Side view of the setup. The viewports at the
top and the bottom of the vacuum chamber are mounted under an angle of 60◦ with respect
to the equatorial plane. Additional beams for Doppler cooling and ion state manipulation
are guided through these windows. The single coil on top of the vacuum chamber is used to
compensate the earth’s magnetic field.
can be minimized by compensating for the stray electric fields and shifting the ion back into the
potential node. This can be done by applying small voltages to the compensation electrodes,
ranging typically from 1 V to 150 V.
4.2. Vacuum vessel
The vacuum chamber housing the ion trap is a stainless steel octagon with eight CF63 flanges.
Three of the eight flanges are equipped with inverted viewports5 the rest have regular viewports
attached6 . The inverted viewports allow us to place custom made lenses7 close to the ions and to
steer or replace them without opening the vacuum chamber. The proximity to the ions guarantees a
good imaging resolution, a high photon collection efficiency and the ability to tightly focus lasers to
address individual ions. The top and the bottom of the chamber consist of two CF200 flanges each
carrying two CF40 flanges with windows mounted under an angle of 60◦ relative to the equatorial
plane. Additional CF16 feedthroughs are attached to the bottom and top flange for wiring the ion
trap and the calcium oven. A schematic drawing can be seen in figure 4.3a.
Not shown in this drawing are the ion getter pump8 , the titanium sublimation pump9 and
the Bayard-Alpert-Gauge10 attached to the chamber via a six way cross. The ion pump runs
5 Ukaea,
fused silica, anti reflection coating (Tafelmaier, 397nm and 720-870nm on vacuum side only)
fused silica, anti reflection coating (Tafelmaier, 397nm and 720-870nm)
7 Silloptics, Germany
8 Varian Star Cell, 20 l
9 Varian
10 Varian, UHV-24 Gauge
6 Ukaea,
35
Experimental setup
permanently while the titanium sublimation pump is fired once a week. The pressure in the
chamber stays below the measurement limit of the Bayard-Alpert gauge of 2 · 10−11 mbar.
4.3. Magnetic field coils
Two pairs of magnetic field coils are attached to the vacuum chamber (see figure 4.3a). The coil
pair aligned along the SW-NE direction is generating the quantization field. The distance of the
coils is 300 mm while their inner diameter is 115 mm. This mounting does not quite resemble a
Helmholtz configuration but nevertheless we expect only small magnetic field gradients at the ion
position. Measurements with two entangled ions confirmed that the gradient at the trap center is
smaller than 0.2 G/m. This means that the frequency change, due to this gradient, is smaller than
1 Hz on the qubit transition for two ions separated by 4 µm. The coils are made out of 300 windings
of copper wire. A current of 1 A sent through these coils produces a magnetic field of 3.4 G at the
ion position. The second pair aligned along the SE-NW11 direction is for compensating external
magnetic fields in order to align the quantization axis with the SW-NE direction. An additional
single coil with a 200 mm diameter (see figure 4.3b) is mounted on the top flange to compensate
for magnetic fields in the vertical direction.
All coils are powered by home-built current drivers having a relative current drift of less than
2 · 10−5 in 24 h. This is achieved by actively stabilizing the current by a servo loop. A stable
reference voltage is compared with the voltage drop across a highly stable resistor12 and used as a
feedback signal.
4.4. Optical access and single ion addressing
Several laser beams enter the vacuum chamber from seven different directions (see figure 4.3).
A 397 nm σ + -polarized beam is aligned exactly with the magnetic field direction. It is used for
optical pumping of 40 Ca+ and 43 Ca+ and as an additional Doppler cooling beam for 43 Ca+ . All
laser beams for Doppler cooling, repumping and photoionization are sent through the SE port of
the chamber. The beams are focused by adjusting the output coupler of the fibers. Typical beam
waists are on the order of 100-200 µm. Additional beams for Doppler cooling and repumping
are sent in via the viewports in the bottom of the vessel. These beams are used when a second
photomultiplier tube (PMT)13 at the NW port is added to the primary PMT located at the south
viewport. The only beam that remains operational from the SE corner is the photoionization
beam. In order not to damage the second PMT during loading the light is automatically blocked
by a shutter in front of the PMT whenever the photoionization lasers are on.
Four 729 nm beams for manipulating the qubits are sent onto the ions. The one entering from
the inverted viewport S is used for addressing a single ion out of a chain of ions. This is achieved
by a custom-made lens system14 and additional telescope lenses to widen the beam prior to the
focussing (see figure 4.4(a)). With this setup the expected diffraction limited spot size is 2.9 µm.
11 all
directions are given according to the compass card in figure 4.3
VCS 302
13 Electron tubes, P25C, Quantum efficiency = 28% at 400 nm
14 Sill Optics, Germany, antireflection coated for 397 nm and 729 nm
12 Vishay,
36
4.5 Fluorescence detection
Figure 4.4.: (a) Custom-made lens setup for focusing a 729 nm laser down to < 4 µm
for addressing a single ion. The same lens system together with a dichroic mirror is used to
image the ion string onto the PMT and the camera. The objective covers about 2.5 % of
the full solid angle. A dichroic mirror is used to separate the incoming light at 729 nm from
the fluorescence light at 397 nm. (b) Measurement of the intensity profile of the laser beam
at 729 nm entering from the south port of the vacuum chamber. For this measurement two
ions where first shifted in position with respect to the laser beam and then excited by a
laser pulse. The pulse length was chosen such that the ion undergoes 3π rabi flops when the
beam is centered on it. The intensity of the beam was then calculated with the recorded
excitation.
Two ions were used to probe the experimentally achieved width of the beam by moving them
along the trap axis and exciting them with a laser beam (see figure 4.4(b)). The pulse length and
intensity of the laser pulse were chosen such that the ion undergoes a 3π rotation when the beam
is centered on it. The spatial intensity profile of the beam was then deduced from the recorded
excitation to the D state. This measurement revealed a FWHM of the beam of 2.6 µm.
Two beams coming from the directions NE and NW are focussed to about 20 µm. The NE
beam is σ + polarized for driving the S1/2 , mF = 1/2 to D5/2 , mF = 3/2 transition with maximal
efficiency. The NW beam is linearly polarized for driving ∆m = ±2 transitions with maximal
efficiency. The beam size is a compromise between maximum intensity and the requirement to
illuminate up to three ions equally. Furthermore, the ions should not reside at a position where
the intensity profile has a step gradient to minimize coupling strength fluctuations due to beam
steering effects. The two beams are not operated simultaneously. In the experiment we can switch
from one to the other by exchanging the fiber after the switching AOM (see figure 4.7 AO3).
Another 729 nm beam enters the vacuum chamber from a viewport in the bottom of the chamber
under an angle of 60◦ with respect to the equatorial plane. This beam is only focused with the
output coupler of the fiber to about 150 µm. It is used to illuminate all trapped ions equally.
4.5. Fluorescence detection
To collect the fluorescence light at 397 nm we again utilize the custom made lens design used for
focusing the beams at 729 nm (see figure 4.4(a)). It is a five-lens objective which is antireflection
coated for 397 nm and 729 nm. The abberations introduced by the fused silica window of the
inverted viewport are corrected by the lens-design. The entrance diameter of the objective is
d=38 mm and the distance to the ions from the first lens is r=58 mm. The photon collection
37
Experimental setup
Figure 4.5.: (a) Fluorescence histogram for two ions each in a superposition of S and D.
In this plot laser intensity and detection time (3 ms) where chosen such that a detection
fidelity exceeding 99.9% was achieved. Two thresholds are used to discriminate between
three different two ion states. Either both ions are in the D state, both are in the S state
or one is in the D state and the other in the S state with a respective mean photon count
rate of 30, 270, 510 photons in 3 ms. The probabilities of finding k bright ions are denoted
pk . (b) The histogram illustrates the discrimination of the two-qubit states S1/2 and D5/2
for a single ion using two PMTs. For each data point the fluorescence of the ion, prepared
in a superposition of the S and D state, was measured for 250 µs. On average we acquire
0.07 photons during this detection time for the ion in the D state. If the ion is in the S
state 7.8 photons are detected. By introducing a threshold between one and two detected
photons one can discriminate between the two states. The intensity of the detection laser
was chosen such that the ion was not heated beyond the mean phonon number achieved
with Doppler cooling. Additionally the time of the detection was made as short as possible
to shorten the duration of the sequence. Despite these restrictions a detection infidelity
of wrongly assigning a quantum state of 0.32 % was achieved. For an application of this
detection technique see chapter 7.
efficiency is given by the solid angle of the objective which is
dΩ
1
=
4π
2
Ãs
1
1−
2
1 + ( 2r
d )
!
≈
1
.
40
(4.2)
Additional losses of 4% arise due to absorption in the objective and another 6% by a narrow
bandpass filter15 used to suppress all light outside the wavelength range 393 nm to 397 nm. In the
present setup we image the ions on two PMT’s whose outputs are combined with a fast adder and
then counted with a multi-purpose counter card16 . In one of the imaging channels a switching box
(mirror, 90:10 beamsplitter, coated window) allows us to send all light to the PMT, to distribute it
between a CCD-camera and the PMT or to send all light to the camera. In front of both PMT’s,
at the location of the image, a variable slit aperture17 is installed to suppress stray light. The
imaging plane is about 1.5 m behind the objective which results in a magnification of 24.5. For all
experiments conducted in this thesis, unless mentioned otherwise, state detection was performed
with the PMT.
The quantum state of a single ion is detected with a PMT in the following way. For a given
15 PMT
II : Semrock FF01-390/18-25; PMT I : Semrock FF01-377/50-23.7-D
Instruments, PCI 6711
17 Owis, Spalt 40
16 National
38
4.5 Fluorescence detection
amount of time, typically 3 ms, the fluorescence lasers at 379 nm and 866 nm are switched on
and all photons detected during that time are added up by a counter card18 . If the number of
detected photons is below a certain threshold we assign the D5/2 state to the result, if it is above
the threshold we assign the S1/2 state. The number of events above/below the threshold relative
to the total amount of repetitions determines the probabilities p1 and p0 corresponding to |α|2
and |β|2 of equation (2.1). This state discrimination also works for multiple ions by setting k
thresholds and determining the pk probabilities of k fluorescing ions. A typical histogram for the
counts acquired for two ions in 3 ms with 3000 repetitions for an equal superposition of S1/2 and
D5/2 can be seen in figure 4.5(a).
For the experiments conducted in chapter 7 it was important that the ions are not heated during
detection as additional entangling operations had to be carried out after the state detection. Up
to three state detections were required during one experiment. Thus a short detection time was
essential in order to prevent a loss of phase coherence of the unmeasured quantum bits due to
decohering processes occurring during the detection interval. Finally the detection fidelity had
to stay above 99%. In order to fulfill all requirements we installed a second PMT and added its
output to the first one and switched off/shielded all light sources in the lab. With this effort we
were able to detect on average 7.8 photons within 250 µs at a dark count rate of 0.07 when using
a 397 nm laser that had its power and detuning set as for Doppler cooling. This ensured that
the ions stayed at the Doppler limit during the detection. The signal to noise ratio >100 enabled
us to get low conditional probabilities for wrong quantum state assignments. The probability of
detecting 0 or 1 photons even though the ion was in the bright state, was p(D|S) =0.24%. The
probability of detecting more than one photon was p(S|D) =0.39% if the ion was in the dark state.
The second device used in some experiments for detecting the quantum state of the ions is an
Electron Multiplying Charge Coupled Device (EMCCD)19 camera. The focal plane of the imaging
system is chosen such that the ions are imaged onto the camera chip. For state detection it is
possible to define a region of interest (ROI) around the positions of the ions of typically 10 × 40
pixels for two ions. Only this small region has to be read out, digitized and processed instead
of the full 1024 × 1024 pixels. The spatial information in the fluorescence signal obtained with
the camera enables us to determine not only the number of fluorescing ions but also which ion(s)
is(are) fluorescing. For a proper state discrimination two reference pictures are taken: One with
all ions fluorescing, the other one with switched-off 866 nm laser for a background picture. These
two pictures are subtracted from each other and then all vertical pixels are added up resulting
in a fluorescence image of the ion string that is integrated in the direction perpendicular to the
string. Then a Gaussian is fitted to every single peak. From these fits the 2N different possible
states are being built serving as reference pictures to be compared with the picture of an ion
string whose quantum states are to be measured. To determine the outcome of an experiment all
pictures are treated in a similar way as the reference pictures. First the reference background is
subtracted, then all vertical pixels are added up and the result is multiplied with all 2N reference
states. A maximum likelihood method then determines the outcome of the measurement. With
this method single ion state detection can be done within 5 ms with a fidelity exceeding 99%.
State discrimination with the camera is technically more demanding than the PMT detection as
18 National
19 Andor
Instruments, PCI 6711
iXon DV885JCs-VP, pixel size 8 × 8 µm, quantum efficiency at 397nm 37%
39
Experimental setup
Figure 4.6.: (a) 397 nm laser setup providing the light for Doppler cooling, state detection
and optical pumping. The EOM at 3.2 GHz generates sidebands on the light to address both
of the 43 Ca+ hyperfine ground states.(b) Laser setup for generating the light at 854 nm and
866 nm for repumping the D-states. Both light sources are combined on a 50:50 beamsplitter
and sent with two polarization maintaining fibers to the vacuum vessel. For efficiently
repumping the D3/2 state in 43 Ca+ two additional AOM’s modulate frequencies at 145 MHz
and 245 MHz onto the 866 nm beam.
an additional program is required for controlling the camera and evaluating the pictures taken.
Furthermore this program has to be synchronized with the experiment control software to ensure
proper state detection.
4.6. Laser system
An advantage of calcium ions is that all necessary wavelength can be produced by solid state
lasers. For all dipole transitions commercial diode laser systems from Toptica were employed in
the experiments. The light for driving the S → P transitions was generated by second harmonic
generation of near infrared diode lasers. All diode lasers are stabilized to medium finesse (F=300)
cavities (see Jan Benhelms thesis [94]) with the Pound-Drever-Hall method. The error signal is
fed back onto the piezo of the diode laser grating as well as on the current driving the diode. This
reduces the laser line width to about 100 kHz limited by acoustic vibrations of the cavity mirrors.
The frequency of the diode lasers can be tuned by 3 GHz by changing the cavity length with a
piezo attached to one of the mirrors. The free-spectral range (FSR) of the cavities is 1.5 GHz. The
only laser that is not a diode laser is the narrowband titanium-sapphire (Ti:Sa) laser at 729 nm. It
is stabilized to a high finesse cavity in order to achieve a line width below 10 Hz. The experimental
setup of all laser sources will be described in the following sections.
40
4.6 Laser system
4.6.1. Lasers for Doppler cooling, state detection, optical pumping and
repumping
Laser at 397 nm
For Doppler cooling, state detection, optical pumping and repumping the ions are excited on the
397 nm dipole transition from S1/2 to D5/2 (see figure 2.3 and figure 2.4). This light is generated by
frequency doubling a diode laser system20 running at 794 nm. At the output of the doubling stage
we have between 5 and 10 mW of blue light which is split by a polarizing beamsplitter (PBS) into
a σ beam and a π beam (see figure 4.6(a)). Both beams are switched on and off by acousto-optical
modulators (AOM)21 driven at 220 MHz. In the σ beam line there is an additional electro optic
modulator (EOM)22 creating sidebands on the light. This is necessary to couple both hyperfine
manifolds in the S1/2 ground state of 43 Ca+ to the P1/2 hyperfine states. Both light beams are
sent with polarization maintaining single-mode fibers to the vacuum vessel. The σ beam is sent
onto the ions from the NW viewport with a σ + polarization while the π beam enters the vacuum
vessel either on the SE-viewport or the bottom viewport. An additional frequency-doubled laser
system is available which can be tuned from 393 nm to 397 nm. Currently it is used to generate
a 400 MHz red-detuned beam at 397 nm for Doppler cooling. This beam is needed to efficiently
recrystalize two ions after a melting of the ion crystal e.g. due to a background collision. It is
superimposed with the σ beam on a PBS having a 90◦ rotated polarization.
Repumping lasers
For repumping the ion from the D3/2 and P1/2 states light at 854 nm and 866 nm is needed. This
light is provided by two diode laser systems23 . The 854 nm laser is switched on and off by an
80 MHz AOM in double pass configuration to ensure proper switch-off(see figure 4.6(b)) of the
laser beam. The 866 nm laser is switched by a single pass 80 MHz AOM. AOM’s at 145 MHz
and 245 MHz create additional light beams that are recombined at beamsplitters for efficiently
repumping 43 Ca+ . This increases the Doppler cooling efficiency and the fluorescence signal. These
AOM’s are only used when running the experiment with 43 Ca+ ions. The light beams of both
lasers are superimposed on a 50:50 beamsplitter and then sent to the SE and the bottom viewport
of the vacuum vessel by two polarization maintaining single-mode fibers.
4.6.2. Ultra-stable titanium-sapphire laser at 729 nm
With a laser at a wavelength of 729 nm we have the capability of exciting the ions on the S1/2 to
D5/2 quadrupole transition. Such a laser offers multiple applications, like:
• spectroscopy on the quadrupole transition
• sideband cooling to the motional ground state
• frequency resolved optical pumping
20 Toptica
DL-SHG
Technology, 3220-120
22 New Focus, 4431
23 Toptica, DL100
21 Crystal
41
Experimental setup
Figure 4.7.: Laser setup to provide 3 different 729 nm light beams and the light used for
stabilizing the laser onto a high finesse cavity. The light intensity of the Ti:Sa is stabilized
by feeding back the signal of PD1 to the RF power driving AO1. AO2 is used for intensity,
frequency and phase manipulation of the light sent to the ions. AO3 to 5 are used as switches
to guide the light through fibers to different ports of the vacuum chamber. The feedback
to the Ti:Sa using a Pound Drever Hall scheme is threefold. A low frequency feedback onto
the tweeter and the Brewster plate as well as a mid and a high frequency feedback to an
intracavity EOM is applied.
• coherent manipulation of optical qubits
• optical shelving for state detection
• state transfer and initialization
These tasks are very demanding regarding the laser’s frequency and power stability. Furthermore
it necessary to tune the laser over 100 MHz within a microsecond and set it to different transitions
for 43 Ca+ and 40 Ca+ which have a frequency difference of 5.5 GHz. The setup of the laser and its
performance is described in [94] and [118]. Here the idea of the locking scheme is recapitulated
and the optical setup is described.
To meet all requirements listed above, the laser is stabilized to a vertically mounted [119] high
finesse cavity24 (F = 412 000, line width = 4.7 kHz, FSR = 2 GHz) made out of ultra low expansion
material (ULE). It rests on teflon feet inside a temperature stabilized (± 1 mK) vacuum tank
(10−8 mbar). The vacuum tank is inside a temperature stabilized wooden box shielding the cavity
from acoustical perturbations. Typical drift rates of the cavity measured by referencing it to the
ions (see chapter 5) are 3 Hz/s or less which means that the cavity spacer length changes within
a year by no more than 20 nm.
This high stability comes at the price of being unable to tune the cavity as both mirrors are
optically contacted to the ULE spacer. In order to get the necessary tuning range we use an AOM25
with a center frequency of 1.5 GHz and a tuning range of 1 GHz in double-pass configuration. With
24 Advanced
25 Brimrose
Thin Films, CO, USA
GPF-1500-1000
42
4.6 Laser system
this AOM, located between the laser output and the high finesse cavity (see figure 4.7 AO6), we
are thus able to tune the laser to any desired frequency. The AOM also allows us to cancel the
drift of the cavity by comparing the laser frequency with the ions transition frequency and feeding
back on the frequency of the AOM.
The optical setup and the servo loop for the Ti:Sa laser are shown in figure 4.7. The feedback
signal is generated by a Pound-Drever-Hall (PDH) [120] locking scheme. An EOM26 modulates
sidebands onto the light and the light reflected by the cavity creates the error signal detected by
a photodiode (P4). The locking scheme of the laser consists of three parts. Slow fluctuations
are actively canceled by sending the error signal through a PI-servo and applying it to the piezo
(tweeter) moving one of the Ti:Sa cavity mirrors and the scanning brewster plate. This loop
has a servo bandwidth of 10 kHz limited by the mechanical resonances of the piezo. A higher
servo loop bandwidth is achieved by inserting an EOM inside the laser cavity. By changing the
refractive index of the EOM the effective cavity length is stabilized. On one electrode of the EOM
we apply the error signal sent through a fast proportional amplifier27 (10 MHz bandwidth). The
other electrode is connected to a high voltage amplifier28 (up to 400 V) which receives as an input
the error signal sent through another PI-servo. The grounds of the two amplifiers are connected,
thus the EOM is floating. With this setup we get servo bandwidths of 300 kHz for the high voltage
part and 1.6 MHz for the fast amplifier.
The laser and the high finesse cavity are coupled by a polarization maintaining single-mode fiber.
To avoid a frequency broadening of the light transmitted through the fiber by acoustic noise, we
implemented an active fiber noise cancelation [118, 121]. The light passing the fiber is first sent
through a 50:50 beam splitter. One part is retro-reflected by a mirror and sent onto a photodiode.
The second part is sent through an 80 MHz AOM (see figure 4.7 AO7) and then through the fiber.
The end-facet of the fiber is at right angle, thus 4% of the light intensity are back-reflected. The
back-reflected light again passes the AOM and hits the same photodiode as the incoming beam.
The beat note at 160 MHz contains the phase fluctuation imprinted on the light by the fiber. An
error signal is created by comparing the beat note to a stable frequency source29 . By implementing
a phase locked loop, the frequency of the beat signal is kept in phase by feeding back onto a voltage
controlled oscillator providing the input signal of the AOM. The light power sent through the fiber
is power-stabilized by monitoring the power transmitted through the high finesse cavity. In this
way beam pointing and polarization variations are also canceled such that the PDH error signal is
not affected.
For the experiments described in chapters 6 to 8 the line width of the 729 nm laser and its spectral
purity plays a crucial role. To determine these quantities we performed a beat measurement with
a similar laser located at another institute 500 m away. A fiber was used to transmit the light
from one building to the other. To avoid any line width broadening of the light we employed a
fiber noise cancelation. The slow relative drift of the lasers was canceled by applying a continuous
slow frequency change to AO7. The spectrum of the beat signal at a center frequency of 10.8 MHz
was recorded with a spectrum analyzer. Figure 4.8(a) shows the power spectrum of the beat for
26 Linos/Gsänger
PM25
HVA-10M-60-F
28 base on Apex, PA98
29 Rhode & Schwarz, SML01
27 Femto
43
Experimental setup
Figure 4.8.: (a) Beat measurement of two 729 nm laser connected via a 500 m long noise
stabilized fiber. A Lorentzian fit yields a FWHM line width of 1.8 Hz within a measurement
interval of 4 s. The inset shows the result of Ramsey experiments on the magnetic field
insensitive transition S1/2 , F = 4, mF = 4 → D5/2 , F = 4, mF = 3. of 43 Ca+ . The contrast
of the Ramsey experiments is plotted versus the Ramsey waiting time. The red line is a
Gaussian fit to the date with a FWHM of 8.1 ms reflecting the loss of coherence due to the
finite laser line width and additional frequency noise introduced by a 2 m fiber guiding the
light to the ions.(b) Power spectral density of the laser beat with a resolution bandwidth of
1 kHz where the reference laser was spectrally cleaned with a filter cavity. The transmission
curve of the filter cavity is given by the blue dashed line. The measurement is limited by
the dynamic range of the spectrum analyzer and the reference laser as can be seen from the
normalized spectrum of the error signal. An integration of the power spectrum shows that
a fraction of less than 10−4 of laser light power is outside a ±250 Hz window around the
laser’s carrier frequency.
a 4 s acquisition time and a resolution bandwidth of 1 Hz. A Lorentzian fit yields a FWHM line
width of 1.8 Hz. If we assume both lasers have the same Lorentzian frequency spectrum we get a
line width for each laser of 0.9 Hz.
To get a better picture of the spectral features of our laser, the spectrum of the second laser was
cleaned with a filter cavity. The transmission function of this cavity is given in figure 4.8(b) together
with the normalized power spectral density of the beat measurement for a resolution bandwidth
of 1 kHz. The figure also shows the spectrum of the laser lock error signal. By comparing both
spectra we can easily identify the characteristic humps in the spectrum as “servo bumps“ of the
laser. Furthermore we can conclude that the beat measurement is limited by the reference laser
and the dynamical range of the spectrum analyzer. An integration of the power spectrum shows
that a fraction of less than 10−4 of laser light power is outside a ±250 Hz window around the
laser’s carrier frequency.
For time scales larger than the 4 s used to determine the spectrum the beat is dominated by
an oscillatory frequency variation with a few seconds period. This accumulates to an effective
line width of 50 Hz in a 4 h acquisition time. This is also consistent with Ramsey measurements
performed on the magnetic field insensitive transition S1/2 , F = 4, mF = 4 → D5/2 , F = 4, mF = 3.
of 43 Ca+ (see figure 4.8(a) inset). From the Gaussian fit to the data we determine a FWHM decay
time τ1/2 of the Ramsey contrast of 8.1 ms. This decay time can be directly transferred into an
44
4.7 Computer control and RF generation
Figure 4.9.: Sketch showing the interconnection of different hardware components. The
experiment is controlled by a LabView programm running on a windows computer (main
PC). Additionally a computer controlling the camera and performing automatized data
evaluation is connect via a TCP/IP connection. The main computer controls the remaining
hardware, reads out the counter data of the PMT and programs the versatile frequency
source (VFS). The VFS is triggered by the AC power line and provides TTL signals and all
phase-synchronous radio frequency signals.
effective laser line width with [122]
∆ν =
2 ln 2 1
.
π τ1/2
(4.3)
From this formula we can infer a laser line width on the ions of 55 Hz for a measurement time of
10 Minutes.
The output of the laser is intensity stabilized by monitoring a small portion of the light by
a photodiode30 (PD1), comparing it with a stable voltage reference and feeding back the error
signal onto the RF power driving AO1. The frequency and intensity of the light sent to the ions is
controlled by AO2. We can choose between three different fiber ports which are switched on and
off by AO3 to 5. AO3 can be driven simultaneously by two frequencies to create a bichromatic
light field. Typically the frequency difference between the two beams is 2.4 MHz which leads to a
diffraction into slightly different directions with an angular separation as small as 0.025◦ such that
the coupling efficiency to the single mode fiber is reduced by about 15% compared with a single
frequency beam.
4.7. Computer control and RF generation
Qubits in ion trap quantum information experiments are manipulated by laser pulses whose power
and length have to be controlled. Furthermore phase coherence has to be ensured for all pulses.
This can be achieved by using AOMs which directly transfer the frequency, power and phase
information from the radio frequency onto the light. In this way, the problem is reduced to
generating precisely timed digital signals and phase-coherent radio frequency pulses with adjustable
length and amplitude. For this purpose a versatile frequency source [105, 123] (VFS) was developed.
It is based on a field programmable gate array (FPGA) which controls a direct digital synthesis
(DDS) board. This DDS board can generate pulses with 16 different frequencies in the range of
30 Thorlabs
PDA 100A-EC
45
Experimental setup
0-300 MHz and allows for phase coherent switching between them. The pulses have a minimal
duration of 10 ns but are typically in the range of 1 µs to 1 ms. The amplitude and shape of
the pulses are controlled by a variable gain amplifier. The frequency range of the DDS board is
limited by filters to 300 MHz. Higher frequencies like the 3.2 GHz for driving the hyperfine qubit
are created by appropriate mixing and filtering of the signals. The frequency resolution is about
0.1 Hz. Additionally up to 16 logic channels (TTL) can be used which are also set by the FPGA
controller. These TTLs are used to control all fast and precisely timed switching required during
an experimental cycle. Logic signals can be sent to the FPGA via 8 input channels.
The FPGA is connected to a PC (main PC) by an ethernet connection and is programmed via a
python31 server. Figure 4.9 shows a sketch of the interconnections between the different hardware
components. The software running on the main PC controlling the experiment is written in
LabView. This program is used to send command sequences to the FPGA board which are executed
as soon as it receives a trigger on one of its input channels. Sequences run synchronously with the
50 Hz power line cycle to avoid varying magnetic fields from one execution to the next. This is
again insured by a trigger to one of the input channels. The LabView program also controls several
National instrument cards32 . They are used to generate analog signals for controlling the power
and frequency of the dipole lasers. TTL outputs of these cards are used to switch time-noncritical
signals.
The PMT signals acquired by the counter card during an experiment are read out, processed and
displayed by the LabView program. The program can be used to run automatized sequences. The
data acquired during such sequences can be sent to the camera PC for data evaluation and model
fitting. The values determined by these procedures are then fed back into the LabView program.
A typical application consists in calibrating the duration of all pulses required in a sequence.
Another LabView program running on the camera PC controls the camera hardware and is
responsible for read-out and evaluation of the camera pictures. The programs on both computers
are communicating via ethernet.
Signal generators are computer controlled by a GPIB-Bus33 . The ISEG high voltage source
providing the tip voltage is connected by a CAN-bus34 to the control PC.
To ensure phase-coherence between all RF-signals the signal generators as well as the VFS-box
are referenced to a 10 MHz GPS assisted quartz oscillator35 .
31 High-level
programming language
Instruments PCI 6711, PCI 6703, DIO 64
33 IEEE-488
34 Controller area network, serial bus standard
35 Menlo Systems GPS 6-12
32 National
46
5. Experimental techniques
This chapter will outline techniques used on a daily basis in the laboratory. It will explain how to
load ions into the trap and detail the structure of a typical experimental sequence. An important
point is the referencing of the 729 nm laser to the ions, which is one of the key elements for
doing high precision spectroscopy and QIP experiments. Then implementation of arbitrary qubit
operations will be demonstrated with the help of ion shuttling and the capability of single ion
addressing. How to analyze the state of an ion by state tomography will be presented at the end
of this chapter.
5.1. Loading ions by photoionization
Typically, ions are loaded by photoionization of an atomic beam produced by an oven. The oven
consists of a stainless steel tube which is heated by an electric current. The tube is about 8 cm
long to ensure a proper collimation of the beam through the trap center, thus avoiding excessive
deposition of calcium on the trap electrodes. The photoionization beam is sent through the trap
center at a slightly different angle. The ions are directly ionized inside the trap volume. Compared
to loading by electron impact it has several advantages. Electron bombardment creates patch
charges on the trap which have to be compensated in order to avoid micromotion. It requires a
higher atomic flux to get a decent loading rate because this ionization process is five orders of
magnitude smaller as compared to photoionization. This leads to bigger calcium layers on the trap
surfaces which increases the patch charge problem. Furthermore, electron bombardment is not
isotope selective and even atoms from the background gas can be ionized. Especially this point is
of importance as we want to load 40 Ca+ and 43 Ca+ deterministically.
Calcium is photoionized by a two step process (see figure 5.1(a)). A laser in Littrow configuration
at 423 nm resonantly excites the atom from the 4s1 S1 to the 4p1 P1 state. From there it can be
ionized by a second step using light with a wavelength smaller than 390 nm. In the experiment
we use a free-running laser diode at 375 nm. Both beams are superimposed on a beamsplitter
cube and sent through a single-mode fiber towards the ion trap via the SE-viewport of the vacuum
chamber (see figure 4.3). Typically 50 to 100 µW for the 423 nm laser and about 500 µW for the
375 nm laser are used for efficient loading.
Isotope selective loading can be achieved by tuning the 423 nm laser to the 4s1 S0 ↔ 4p1 P1
resonance (see figure 5.1(b)) of the respective isotope. The frequency of the 423 nm laser can be
determined either on the wavemeter or by saturation spectroscopy on a calcium vapor cell. The
spectroscopy clearly shows the Doppler-free resonance for the most abundant calcium isotope 40 Ca.
By detuning the laser by 612 MHz to the blue it becomes resonant with the
43
Ca resonance. Due
to the finite Lorentzian line shape and the natural abundance of calcium the relative loading rate of
47
Experimental techniques
Figure 5.1.: (a) Two step photoionization of calcium. The 423 nm laser can be used to
make the process isotope selective. (b) Excitation spectrum of calcium at 423 nm. Figure is
taken from [124]. The numbers above the curve show the resonances of the respective calcium
isotopes. 43 Ca+ has an isotope shift of 612 MHz with respect to the 40 Ca+ resonance. The
red line is fit to the measured data taking into account a single resonance for 40 Ca+ . The
residuals between the red and the black curve are shown in green and indicate the resonances
of other isotopes.
43
Ca+ compared to other isotopes is only 50%. To further increase the isotope selectivity there are
two ovens in the apparatus. One is filled with calcium granules with a natural isotope abundance
of 97% 40 Ca and the other is filled with an enriched source consisting of 81% 43 Ca, 13% 40 Ca and
5% 44 Ca.
For loading calcium ions the Doppler cooling laser at 397 nm, the repumper at 866 nm and both
photo-ionization lasers are continuously on. The current in the oven is set such that about two
ions are loaded per minute. By monitoring the fluorescence signal with the PMT and/or camera,
ions appearing in the trap can be detected. Larger crystals are loaded by waiting for longer times.
In case that too many ions are in the trap all of them have to be discarded by switching off the
RF drive. Then the loading procedure is started again.
After the intended number of ions has been loaded, the Doppler cooling lasers are adjusted for
optimum detuning and power [92]. Two different power settings for the 397 nm laser have to be
optimized: One for proper Doppler cooling, to get as close as possible to the Doppler-limit and
the other for a maximum fluorescence to get a good state detection. The parameters for the two
settings will be given in the following section. When all dipole lasers are properly set we can start
running experimental sequences manipulating the qubit.
5.2. Experimental sequence
All experiments are executed in a pulsed fashion. Such a sequence typically consists of six building
blocks (see figure 5.2) which are described in the following. A single experiment is typically repeated
50-200 times.
48
5.2 Experimental sequence
Figure 5.2.: Typical sequence of laser pulses executed in an experimental cycle. (1) Doppler
cooling, (2) optical pumping, (3) sideband cooling, (4) frequency resolved optical pumping,
(5) quantum state engineering, (6) state detection. Each sequence is triggered by a TTL
pulse (0) synchronized with the line cycle and is typically repeated 50, 100 or 200 times.
(1) Doppler cooling
For 40 Ca+ , it is sufficient for efficient Doppler cooling to use a single 397 nm beam coupling
S1/2 → P1/2 and a single 866 nm laser repumping the transition D3/2 → P1/2 . To cool all three
normal modes of motion of the ion in the trap, care has to be taken that the Doppler-cooling
beam has sufficient overlap with all modes. The power of the 397 nm laser is set such, that its
Rabi frequency is half the saturation intensity [92] of the dipole transition. It is red detuned with
respect to the resonance by half the linewidth of the transition. The 866 nm laser power is set such
that the fluorescence is just below saturation to avoid additional line broadening. It is slightly blue
detuned to avoid coherent population trapping. The degeneracy of the Zeeman sublevels is lifted
by a magnetic field.
For 43 Ca+ Doppler cooling is a little bit more complicated. Here three light fields for the
397 nm laser are necessary. Two of the light fields are σ + - polarized, where one is coupling
the S1/2 , F = 4 → P1/2 , F = 4 transition and the other one, detuned by 3.2 GHz, is coupling the
S1/2 , F = 3 → P1/2 , F = 4 transition. The 3.2 GHz component is created by modulating sidebands
on the light field with an EOM (see figure 4.6). The third light field is a π polarized beam which
is needed to avoid pumping into the dark state S1/2 , F = 4, mF = 4.
In the experiment it turned out that the π and the σ + beam should not have the same frequency.
If they do, Doppler cooling is not working properly which might be due to dark states that are
inefficiently cooled. A detuning of 5 MHz to the red for the π beam, with respect to the σ + light
field, turned out to yield the best Doppler cooling results. The 866 nm laser is tuned close to the
D3/2 , F = 3 → P1/2 , F = 3 transition. For efficient repumping additional light fields shifted by
-145 MHz and -395 MHz are provided (see figure 4.6). In this way all D3/2 hyperfine levels are
coupled to one of the P1/2 states. Power and detuning from the resonances for both lasers are set
the same way as for 40 Ca+ . A magnetic field lifts again the degeneracy of the Zeeman states.
The fluorescence collected during Doppler cooling is used to check if the ions are in a crystal-like
structure. If the fluorescence collected for a single sequence falls below a certain threshold the
whole experiment is discarded and executed again.
49
(2) Optical pumping
A weak short pulse with a σ + polarized 397 nm laser together with an 866 nm laser pulse is used
for optically pumping the ions into either S1/2 , mj = 1/2 for 40 Ca+ or S1/2 , F = 4, mF = 4 for
43
Ca+ . As for Doppler cooling, more light fields have to be used for pumping 43 Ca+ . After this
step we achieve a pumping fidelity of 99% into the desired state within 70 µs for 40 Ca+ and about
98% for 43 Ca+ ions. The pumping fidelities are limited by beam polarization and overlap of the k
vector of the light field with the magnetic field axis.
(3) Sideband cooling
Sideband cooling to the motional ground state is done in the same way for both isotopes. The
729 nm laser is tuned to the red sideband of the stretched state S1/2 , mj = 1/2 → D5/2 , mj = 5/2
for 40 Ca+ and S1/2 , F = 4, mF = 4 → D5/2 , F = 6, mF = 6 for 43 Ca+ . Every state transfer from
S1/2 to D5/2 reduces the mean phonon number by one quantum. To reduce the lifetime of the
D5/2 level a 854 nm laser is switched on at the same time. The population is transferred from the
D5/2 state to the P3/2 state from where it decays back to the ground state. To avoid population
leaking from the cooling cycle due to a decay into D3/2 , an 866 nm laser is continuously on. This
repumping via the P1/2 state does not guarantee a decay into the desired S1/2 state. Additional
optical pumping with the 397 nm laser brings the population back into the cooling cycle. After
about 7 ms a mean phonon number of hni = 0.05(5) is reached.
(4) Frequency resolved optical pumping
To further increase the optical pumping efficiency we make use of the 729 nm laser. The scheme
for 40 Ca+ is similar to sideband cooling. For a duration of 500 µs the 729 nm laser is switched on
together with the 866 nm and 854 nm laser. The quadrupole laser couples the states S1/2 , mj =
−1/2 → D5/2 , mj = 3/2 depleting the undesired Zeeman state. Due to the repumping the population piles up in the S1/2 , mj = 1/2 and we get pumping efficiencies exceeding 99.8% for a single
ion.
In 43 Ca+ we transfer the population in S1/2 , F = 4, mF = 4 to the D5/2 , F = 6, mF = 6
state. Then we again apply an optical pumping step with the 397 nm laser. This redistributes
the population remaining in the ground state. Another π pulse exchanges the populations in
S1/2 , F = 4, mF = 4 and D5/2 , F = 6, mF = 6. After this step 98% of the population should
be in S1/2 , F = 4, mF = 4 and the rest in D5/2 , F = 6, mF = 6. A light pulse with 854 nm
laser clears out the D5/2 state via the P3/2 , F = 5, mF = 5 from where it can only decay into
S1/2 , F = 4, mF = 4. With this procedure we achieve pumping fidelities for a single
43
Ca+ ion of
more than 99.4%.
(5) Quantum state manipulation
After the ions are prepared in the motional ground state and the desired electronic state, the qubit
in
40
Ca+ and
43
Ca+ can be manipulated by 729 nm laser pulses. If the qubit is encoded in the
hyperfine states of
43
Ca+ it can also be manipulated by microwave pulses.
50
5.3 Referencing the 729 nm laser to the ions
Figure 5.3.: Variation of the magnetic field over the line cycle. The data points were
measured by delaying the start of a Ramsey experiment with respect to the line trigger.
The strong 50 Hz component is most likely caused by transformers and other electronics of
the setup.
(6) State discrimination
By detecting the fluorescence of the ion on either the camera or the PMT one can distinguish
between the S1/2 level and the D5/2 level. The power of the 397 nm laser is increased for a high
fluorescence rate to get a good state discrimination within a few milliseconds. To discriminate
between the states of the hyperfine qubit S1/2 , F = 4, mF = 0 and S1/2 , F = 3, mF = 0, additional
transfer pulses are used to hide the S1/2 , F = 4, mF = 0 population in the D5/2 manifold. Two
transfer pulses to different D5/2 Zeeman states ensure that the detection efficiency is above 99%.
5.3. Referencing the 729 nm laser to the ions
In order to make use of the 729 nm laser for QIP experiments an exact knowledge of the magnetic
field and the laser frequency is necessary. Therefore the laser is referenced to the ions in the trap.
The referencing is done by probing two different transitions (a,b) in either 40 Ca+ or 43 Ca+ . From
the exact knowledge of the Zeeman splitting of the transitions one can then infer the relative laser
frequency and the magnetic field. By recording the transitions for several minutes it is possible
to determine the drift of the reference cavity and, by active feed forward, to stabilize the output
frequency of the laser up to some small fluctuations caused by magnetic field variations.
The ions are Doppler-cooled and initialized by optical pumping. Then a Ramsey type scheme [125]
is employed to determine the transition frequency. A π/2 pulse creates a superposition of the two
levels to be probed. Then after a Ramsey waiting time τR , a second π/2 pulse is applied. One transition is probed twice in this way. The first time the second pulse has a relative phase of φ1 = π/2
and the second time it has a phase of φ2 = 3π/2. After each measurement the population of the
levels is measured as described in chapter 4.5. By repeating this sequence a hundred times one can
infer the probabilities of finding the ion in the excited state pφ1 and pφ2 for the respective phases.
From this one can determine the frequency difference between the laser and the probed transition
as [94]
∆ν/(2π) =
1
pφ − pφ2
arcsin 1
.
2π (τR + 2τπ /π)
pφ1 + pφ2
(5.1)
Where τπ denotes the pulse length to drive a π-pulse on this transition with the chosen laser
51
power. Typical values for τπ are between 5 and 20 µs whereas τR is typically 10 times longer, that
is 0.1-1 ms.
From the deviation of the transition frequencies ∆νa and ∆νb we first determine the magnetic
field. From this knowledge and the frequency of one transition we then determine the transition
frequency at zero magnetic field. This serves as a reference from where all other transitions
frequencies can be calculated for a given magnetic field.
The referencing measurements and the analysis are fully automatized and run about every
minute. The measurements are recorded over time and a polynomial fit to the frequency data
reveals the drift of the cavity. Usually a first order fit is employed and an extrapolation is used to
feedback on the signal generator producing the frequency for the AOM set up between laser and
cavity (see figure 4.7 AO6). With this feedback scheme the laser frequency deviates by less than
200 Hz (rms-deviation) from the transition frequency. The exact value depends on τR which is
limited by the coherence time of the ions, the linearity of the cavity drift and the line width of the
probe laser.
The magnetic field data are averaged over time where data older than 5 min are not taken into
account. The average is used as a a best guess for the magnitude of the magnetic-field during the
next minute. The limiting factor in determining the exact magnetic field is given by erratic jumps
in the magnetic-field by 100-250 µG due to external magnetic fields (elevator etc.). Without these
jumps the magnetic field is stable and can be predicted within about 50 µG over several hours if
the experiments are triggered to the line cycle.
If the time between the start of an experiment and the line trigger is varied, changes of the
magnetic-field by about 1.4 mG are visible (see figure 5.3). To avoid this variation from shot
to shot, experiments are always started at a fixed phase of the power line cycle. Nevertheless,
since some of the transitions used exhibit a magnetic field sensitivity of 2.8 MHz/G frequency
corrections of up to 4.5 kHz have to be taken into account over the course of a sequence that lasts
several milliseconds. To avoid these corrections for short sequences the experiments are triggered
to the maximum in the line cycle (around 250◦ ) such that the magnetic field does not change in
first order. Ways to get rid of this limitation are either technical, by active or passive magnetic
field stabilization, or by choosing a qubit which is insensitive to magnetic field changes [102, 126].
Such a magnetic field insensitive qubit was used in chapter 6 to store entangled states for tens of
milliseconds.
5.4. Implementation of arbitrary qubit operations and ion
shuttling
There are different approaches to carry out single-qubit rotations. A requirement that is always
present is the capability to limit interactions to individual qubits. In the ion trap case this is
equivalent to shining in the laser on one ion at a time. One possibility to achieve this is to use
micro-fabricated segmented traps to manipulate the ion motion [127]. With the help of different
trap electrodes a large ion string can be split into smaller ones and brought to trap zones where
the selected ions interact with laser fields [127]. This approach requires a high degree of control
and stability on the voltages applied to the trap electrodes. Some experiments have demonstrated
52
5.4 Implementation of arbitrary qubit operations and ion shuttling
Figure 5.4.: (a) Laser beam setup for the 729 nm laser. The tightly focused beam 2 is used
for addressed σz rotations. The beams 1 and 3 are for simultaneous operations on up to three
ions. Theses beams are either bichromatic light fields for implementing a MS-interaction or
single frequency for carrier rotations on all ions. The NE beam is σ + polarized for driving
∆m = ±1 transitions with maximal efficiency. The NW beam is linearly polarized for
driving ∆m = ±2 transitions with maximal efficiency. (b) Pulse scheme to perform a π/2
carrier rotation on ion 1 and no rotation on ion 2. Both ions are first rotated by π/4, then
ion 1 gets a π phase shift with the AC-stark pulse. The second π/4 pulse on both ions
rotates ion 2 to the initial state and completes the π/2 rotation on ion 1. (c) Pulse scheme
to perform a MS-gate on two out of three ions. The first π/4 MS-gate partially entangles
all three ions. A stark shift pulse on ion 1 changes the phase of this ion by π. The final π/4
MS-gate entangles ions 2 and 3 and reverts ion 1 to its initial state.
the ability to deterministically split [128, 129] and reorder an ion string [130]. To do this splitting
fast the traps have to be small as the trap frequencies scale as d−9/10 with d the distance of the
ions to the trap electrodes [131]. A problem all traps face is that the heating rate scales as the
d−4 [132, 133] which is especially bad for micro-fabricated traps. Although experiments showed
that cooling the traps [134] reduces the anomalous heating a considerable technical overhead is
required to do QIP with these traps.
Another way to single out ions from a larger crystal is to use strongly focused light beams.
Ideally the focus should be much smaller than the separation of two ions which is typically a few
µm. However, even for perfect optics the Gaussian beam profile will lead to some light on the
neighboring ion and thus some residual coupling Ωres . With our addressing optics for the 729 nm
beam (see figure 4.4) we are able to get the ratio ² = Ωres /Ω for carrier pulses down to about 5%.
This ratio can be further improved by using Stark shift pulses. As the interaction strength for
these scales with the intensity instead of the field strength the addressing error scales now with
²2 . Addressing errors are thus reduced to below 0.3%. One pays for this increased addressing
capability by an increased sensitivity to intensity fluctuations.
(i)
θ
(i)
Single-qubit light-shift gates Uz (θ) = e−i 2 σz are realized by an off-resonant strongly focused
beam impinging (see beam 2 in figure 5.4(a)) on ion i at a 90◦ angle relative to the trap axis. As
the k-vector of this beam is perpendicular to the trap axis this configuration does not allow to
couple to the axial center of mass motion of the ions and can thus not be used to do two-qubit
gate operations. A second beam at a 45◦ angle relative to the trap axis (see beam 1 or beam 3
in figure 5.4(a)), illuminating all ions with equal strength serves to carry out gate operations that
θ
(1)
are symmetric under qubit exchange. Collective single-qubit gates U (θ, φ) = e−i 2 (σφ
53
(2)
+σφ )
, are
Figure 5.5.: (a)Pulse sequence describing a Ramsey experiment on the motional states
of the ion. (b) Ramsey contrast of the experiment versus number of events shuttling the
ion by 10 µm. The inset shows the contrast of a Ramsey phase experiment on the motion
without (•) and with (N) a single ion shuttle.
realized by resonantly exciting the qubit transition and controlling the phase φ of the laser light.
If instead a bichromatic light field is used for implementing a Mølmer-Sørensen interaction, the
(2)
θ (1)
propagator UM S (θ, φ) = e−i 2 σφ ⊗σφ is realized. This interaction is used for entangling ions.
As the tightly focused beam is static we still have to move the ion string to address different
ions. The shuttling of the ions is done in such a way, that the population of the harmonic oscillator
mode does not change during the transport. The curvature of the trapping potential is kept
constant during the movement of the ions. The movement of the ions is achieved by changing
the tip voltage with the circuit presented in chapter 4.1. To avoid energy transfer the ions are
accelerated adiabatically with respect to the trapping frequencies. Typical shuttling times for
10 µm of 40 µs corresponding to about 50 axial oscillations were achieved without affecting the
population of the COM mode. This was confirmed by performing a Ramsey experiment on the two
lowest motional states in the following way: A single ions was cooled to the motional ground state
by sideband cooling. A π/2 pulse on the carrier followed by a π pulse on the red sideband creates
the superposition state |↓, 0i + |↓, 1i. Then the ions were shuttled over a distance of 10 µm along
the trap axis. After a waiting time of 40 µs the ions were shuttled back to the original position.
Again a waiting time of 40 µs ensured that the voltage on the electrodes was settled. The sequence
applied to the ion is shown in figure 5.5(a). The first two pulses were then applied in reverse order
and the phase of the carrier pulse was scanned to record Ramsey fringes (see figure 5.5(b) inset).
The contrast of these fringes was then compared to the case where the ion was not transported. If
the ion was heated during the transport, i.e. the occupation of the motional mode was incoherently
changed, the contrast of the Ramsey fringes decreases with respect to the stationary case. No such
effect could be observed even for moving the ion 32 times (see figure 5.5(b)).
(i)
It can be shown [135] that combining this set of elementary gates {Uz (θ), U (θ, φ), UM S (θ, φ)}
(k)
arbitrary multi-qubit operations and mapping of observables Aij to σz for read-out can be implemented. This is similar to NMR techniques [15] where the spin-spin interaction that is always
present can be switched off by refocusing techniques. The dynamics of the system is thus controlled
by single qubit operations. In the ion trap case, the spin-spin interaction is replaced by the MSgate and the refocusing pulses are implemented with the addressed beam. Two simple examples
for such a pulse sequence with ions and their effective circuit diagram are shown in figure 5.4 b
54
5.5 State tomography
Figure 5.6.: Absolute values of the density matrices for the states (a) |↑↓i+|↓↑iand (b)
|↓↓i+|↑↓ireconstructed by a maximum likelihood method.
and c.
The sequence in figure 5.4 b shows how to do a π2 pulse on only one of two ions. Both ions are
(1)
first rotated by U ( π4 , 0), then ion 1 gets a phase shift with the AC-stark pulse Uz (π). The second
U ( π4 , π) pulse on both ions rotates ion 2 to the initial state and completes the π/2 rotation on ion
1. Example c is similar to the NMR case and is used to entangle two out of three ions. The first
(1)
UM S ( π4 , 0) MS-gate partially entangles all three ions. A stark shift pulse on ion 1 Uz (π) changes
the phase of this ion by π. The final UM S ( π4 , 0) MS-gate entangles ions 2 and 3 and reverts ion 1
to its initial state. So by flipping the phase of one of the ions the interaction is switched off.
5.5. State tomography
State tomography is used to determine the density matrix of a quantum state by measurements on
an ensemble of identically prepared quantum states. As described in chapter 2.1.5, a measurement
of the expectation values λi of the Pauli operators σx , σy , σz is sufficient to directly reconstruct a
density matrix describing a qubit state. For a qubit encoded in the electronic state of a trapped
ion, σz can be directly measured by detecting the fluorescence of the ion (see chapter 4.5). The
other two observables are measured by applying a unitary transformation prior to measuring σz on
the transformed state. This transformation maps the eigenvectors corresponding to the eigenvalues
of σx or σy onto the eigenvalues of σz (see chapter 2.1.3).
For multi-qubit systems it is necessary to measure observables which are tensor products of
Pauli matrices. The measurement procedure requires single-ion addressing and single-ion detection
capabilities. For two ions, observables like σz1 ⊗σy2 are determined by applying an addressed rotation
on qubit 2 prior to measuring σz1 , σz2 (each (±1)) with the CCD-camera. By correlating the results
the expectation value for σz1 ⊗ σy2 can be measured by multiplying the ±1 outcomes and averaging
over all realizations. To reconstruct the density matrix of two qubits nine measurements have to be
ˆ σx , σy , σz }.
carried out to determine the 16 expectation values of all products of σi ⊗σi with σi = {I,
The unitary transformations we apply to the ions are implemented as shown in the preceding
chapter. A unitary transformation to measure e.g. σz1 ⊗ σy2 would exactly look like the sequence
of pulses shown in figure 5.4(b)
³π´
³π ´
Uz1 (π)Ux
.
(5.2)
Ux
4
4
55
A problem one faces is that in an experiment one can never determine the observables exactly,
as an infinite number of measurements is necessary to eliminate the statistical error. Thus a
measurement of an expectation value λ̄i is only a best estimate for the actual expectation value.
This seemingly small difference has important consequences as real physical Bloch vectors have
to lie inside the Bloch sphere. When a state is reconstructed that is close to the boundary the
statistical fluctuations might lead to a single-ion Bloch vector lying outside the unitsphere. In
this case the corresponding density matrix has one eigenvalue > 1 and one eigenvalue < 1. As
this matrix is no longer positive semidefinite it does not describe a physical state. This problem
gets more severe the higher the dimension of the Hilbert-space becomes as more and more density
matrix entries will be close to zero and can easily get negative if not enough data are taken.
This problem can be overcome by making use of the measurement results in another way. The
strategy we use to determine the density matrix is based on a maximum likelihood method [136].
This method searches among all physically possible density matrices the one that is most likely
to reproduce the observed measurement results. The measurements described above project the
quantum state ρex onto a set of different bases. For each of these bases, N copies of ρex were used,
and in Nfi experiments ρex was projected onto the state |Ψi i. The probability of finding exactly
these results for an arbitrary density matrix ρ is given by the likelihood function
L(ρ) =
Y
N fi
hΨj | ρ |Ψj i
.
(5.3)
j
The maximum likelihood estimation for an experimentally obtained density matrix is given by
the state ρ that maximizes the log-likelihood function
L(ρ) = N
X
fj log hΨi | ρ |Ψi i .
(5.4)
j
As L(ρ) is a convex function on a convex set, it has no local maxima which simplifies the task
of maximizing L(ρ). If a directly reconstructed density matrix is valid it will also maximize the
likelihood function, thus both methods reconstruct the same density matrix. Different numerical
methods can be used to maximize the log-likelihood function. The method used to reconstruct all
density matrixes in this thesis relies on an iterative procedure [137] solving a nonlinear operator
equation.
Figure 5.6 shows the absolute values of the density matrices for the states |↑↓i+|↓↑i and
|↓↓i+|↑↓i created in the experiment.
56
6. Entangling calcium ions
The most complicated operations needed for QIP in ion traps are universal multi-qubit operations.
Most implementations so far lack the performance to carry out gate operations in series without
producing too high errors. This chapter will describe experiments on entangling either a pair of
40
Ca+ ions or a pair of 43 Ca+ ions. The first section will focus on a high-fidelity entanglement
of 40 Ca+ ions cooled to the ground state and to thermal states of motion with hni 6= 0. The
entanglement of three 40 Ca+ ions by means of a MS interaction and the implementation of the
first steps to do arbitrary quantum circuits on three ions will be discussed in the next section. The
chapter will be concluded by the description of an experiment where the knowledge gained with
40
Ca+ ions was transferred to the more complicated level structure of 43 Ca+ and used to entangle
a pair of 43 Ca+ ions. Especially interesting is the mapping of the entanglement from the optical
onto the hyperfine qubit.
6.1. High fidelity two ion Bell states
Recently the first application of a Mølmer-Sørensen gate operation to an optical qubit was demonstrated [40]. In this experiment Bell states with a so far unmatched fidelity of 0.993(1) were
deterministically created. Here, a further investigation of this universal gate operation acting on
optical qubits with an extended experimental analysis is presented. Particular emphasis is put on
the compensation of AC-Stark shifts and amplitude pulse shaping to reach high fidelities without
compromising the gate speed substantially. The gate characterization is extended further by investigating the fidelity decay for different input states after up to 21 individual operations. Possible
error sources for the gate operation will be listed at the end.
Moreover, the first experiments demonstrating a universal entangling gate operating on Dopplercooled ions are presented. For ions in a thermal state with n̄ = 18(2), Bell states with a fidelity of
0.984(2) were obtained.
The ability to implement high fidelity multi-qubit operations on Doppler-cooled ions is of practical interest in ion trap quantum information processing as the implementation of quantum algorithms demands several techniques that do not conserve the ions’ vibrational quantum state: (i)
State detection of ancilla qubits as required by quantum error correction schemes [138] can excite
the ion string to a thermal motional state close to the Doppler limit because of the interaction with
the laser inducing the ions to fluoresce. (ii) Experiments with segmented trap structures where
ion strings are split into smaller strings also tend to heat up the ions slightly [139]. Here, the
availability of high-fidelity gate operations even for thermal states may provide a viable alternative
to the technically involved re-cooling techniques using a different ion species [140, 141].
Note that all experiments, except the measurements shown in figure 6.4 demonstrating the high
57
Entangling calcium ions
(a)
(b)
Light power
slope
duration
pulse
duration
Time
Figure 6.1.: Effect of amplitude pulse shaping on non-resonant population transfer caused
by a bichromatic light field non-resonantly exciting the carrier transition. Experimental
results are presented for a gate duration of tgate =25 µs. A comparison of the evolution of
the populations p2 (N), p1 (¦), p0 (•) for a square-shaped pulse (a) with an amplitude-shaped
pulse (b) shows a suppression of the strong non-resonant oscillations for the latter case. The
slopes are shaped as a Blackman window with a duration of 2.5 µs, the figure inset showing
the definitions of pulse and slope duration. Numerical simulations suggest that the actual
pulse shape is not so important as long as the switching occurs sufficiently slowly. The solid
lines are calculated from (3.34) and (3.29). To match experimental data and simulations,
we allowed for a time offset ∆t = 0.5 µs that accounts for the finite switching time of the
AOM controlling the laser power.
fidelity gate on thermal ions, use the S1/2 , mj = 1/2 level together with the D5/2 , mj = 5/2 as a
qubit instead of D5/2 , mj = 3/2. The choice of the qubits was given by the beam geometry and
the requirement of a high coupling strength (see chapter 3.3.1). The center of mass trap frequency
for all experiments, except the measurements shown in figure 6.4 (c) (d) , was set to ωax /(2π) =
1.232 MHz which corresponds to a Lamb-Dicke parameter η = 0.044. For the mentioned exception
the trap frequency was slightly increased to ωax /(2π) = 1.465 MHz which corresponds to a LambDicke η = 0.04. The main findings of this chapter were also published in [76].
Two 40 Ca+ ions were used to perform the experiments in this section. Doppler cooling and
frequency-resolved optical pumping was applied to initialize the ions to |↓↓i. Then the MS interaction is switched on and the appropriate parameters are varied to do the measurements.
6.1.1. Amplitude pulse shaping
The merits of amplitude pulse shaping were studied by observing the time evolution of the populations pk at the beginning of the gate operation when the population transfer is dominated by
fast non-resonant coupling to the carrier. For a better visibility of the effect the ions were cooled
to the ground state. Figure 6.1 (a) shows the population evolution for the first 5 µs of a 25 µs gate
operation based on a rectangular pulse shape. Averaging over a randomly varying phase ζ 1 , strong
oscillations with a period of 2π/δ=0.84 µs were observed. The phase ζ, defined in chapter 3.5,
determines whether the gate operation starts in a maximum or a minimum of the intensity of
1ζ
determines whether the gate operation starts in a maximum (ζ = 0) or a minimum (ζ = π/2) of the intensity
of the amplitude-modulated beam (see chapter 3.5).
58
6.1 High fidelity two ion Bell states
(a)
(b)
Figure 6.2.: (a) Populations p0 (N), p1 (¦), p2 (•) after a single gate operation (tgate =25 µs)
where the global frequency detuning of the bichromatic entangling pulse is varied by scanning
AO 2. A maximally entangled state is achieved for a global frequency detuning of (2π) 37 kHz
relative to the qubit transition frequency due to AC-Stark shifts. (b) Introduction of a beam
imbalance ξ = 0.08 shifts the pattern of the populations by the required amount to fully
compensate for the AC-Stark-shift (note the different x-axis offsets in (a) and (b)). The
solid lines are calculated by solving the Schrödinger equation for the Hamiltonian given in
(3.27) amended by a term accounting for the measured AC-Stark shift.
the amplitude modulated beam. Panel (b) shows that the non-resonant excitations vanish completely after application of amplitude pulse shaping with a slope duration of 2.5 µs corresponding
to three vibrational periods of the center-of-mass mode. The slopes were shaped as a Blackman
window [142], where the form of the shape is chosen such that a shaped and a rectangular pulse
of the same duration have the same pulse area (see inset of panel (b)). Different pulse lengths
are achieved by varying the duration of the central time interval during which the laser power is
constant. The solid lines in the figure are calculated from (3.34) and (3.29).
6.1.2. AC-Stark shift compensation
The AC-Stark shift caused by bichromatic light with spectral components each having a Rabi
frequency of Ω/(2π)=220 kHz (for tgate =25µs) is measured by scanning the global laser frequency
using AO 2 (see figure 4.7). The resulting populations after a gate operation, again for ground
state cooled ions, are depicted in figure 6.2 (a). We observe a drop of the population p1 to zero
at a detuning of (2π) 37 kHz from the carrier transition. At this setting the ions are maximally
entangled. By changing the relative power of the bichromatic field’s frequency components such
that ξ = 0.08 (see chapter 3.5.1) the AC-Stark shift is compensated. This translates the whole
excitation pattern in frequency space as can be seen in figure 6.2 (b).
A more sensitive method to infer the remaining AC-Stark shift δAC after a coarse pre-compensation
consists in concatenating two gates separated by a waiting time τw in a pulse sequence akin to a
Ramsey-type experiment [143] and scanning τw . This procedure maps δAC to a phase φ = δAC τw
which is converted into a population change p2 = cos2 (φ), p0 = sin2 (φ) by the second gate pulse.
For two ions, the corresponding Ramsey pattern displayed in figure 6.3 shows oscillations of the
populations p0 and p2 with a frequency of two times the remaining AC-Stark shift.
59
Waiting time tw (µs)
Figure 6.3.: Population evolution of |↑↑i (•) and |↓↓i (N) when scanning the waiting time
between two 25 µs gate pulses in a Ramsey-like experiment. For this scan the detuning ² was
set to (2π) 40 kHz. In this set of data, the AC-Stark shift was only partially compensated by
imbalancing the power of the two frequency components. From the sinusodial fits shown as
solid lines, we infer an oscillation period of 258(4) µs corresponding to a residual AC-Stark
shift of (2π) 1.94(3) kHz.
6.1.3. Gate analysis
A full characterization of the gate operation could be achieved by quantum process tomography [50].
At present, however, the errors introduced by the tomography pulse and individual qubit detection
are on the few percent level in our experimental setup which renders the detection of small errors
difficult in the entangling operation. Instead, the quality of the gate operation was characterized
by using it for creating different Bell states and determining their fidelities.
For the Bell state Ψ1 = |↓↓i + i |↑↑i, the fidelity is given by F = hΨ1 | ρexp |Ψ1 i = (ρexp
↑↑,↑↑ +
exp
exp
ρexp
describing the experimentally produced state.
↓↓,↓↓ )/2 + Imρ↓↓,↑↑ , with the density matrix ρ
To determine F , the populations p2 + p0 need to be measured at the end of the gate operation as
well as the off-diagonal matrix-element ρexp
↓↓,↑↑ . To determine the latter, a π/2 pulse was applied to
(1) (2)
both ions with an optical phase φ to measure hσz σz i for the resulting state as a function of φ.
This procedure is equivalent to measuring oscillations of the expectation value Tr(P (φ)ρexp ) of the
(1) (2)
parity operator P (φ) = σφ σφ where σφ = σx cos φ + σy sin φ (see figure 6.4 (b) and (d)). The
amplitude A of these oscillations equals 2|ρexp
↓↓,↑↑ | and is obtained by fitting them with the function
Pfit (φ) = A sin(2φ + φ0 ).
Measurements [40] using |↓↓i as input state have demonstrated Bell state fidelities as high as
0.993(1) (see figure 6.4 (a) and (b)) for gate times of 50 µs or 61 trap oscillation periods. Figure
6.4 (a) illustrates the population evolution induced by the gate pulse for ground-state cooled
ions initially prepared in the qubit states |↓↓i. Figure 6.4 (b) displays parity oscillations for the
produced Bell state. By doubling the detuning to ²/(2π)=40 kHz the gate duration is reduced
to only 31 trap oscillation periods and Bell states with a fidelity of 0.971(2) were observed which
is remarkable considering the small Lamb-Dicke parameter of η = 0.044. The detrimental effects
illustrated in figure 6.1 (a) are sufficiently suppressed by amplitude pulse shaping. While many
theoretical papers discussing Mølmer-Sørensen and conditional phase gates put much emphasis on
60
Figure 6.4.: Measured population evolution for p0 (•), p1 (¦), p2 (N) and parity oscillations
with (a,b) and without (c,d) ground state cooling. In the latter case, population is transferred faster into |↑↓, ni, |↓↑, ni as compared to sideband cooled ions due to the higher
coupling strength to the sidebands. In (c), the solid lines are a fit to the data points using (3.36) with the mean phonon number n̄ as a free parameter giving n̄ = 18(2). The parity
oscillations for the ions in a thermal state of motion have an amplitude of 0.980(2). Combining this measurement with the independently determined populations p2 + p0 = 0.988(1)
results in a Bell state fidelity of 0.984(2). The data appearing in (a) and (b) are taken
from [40]. Here the deviation of the dashed lines from the data is caused by the AC-Stark
shift compensation using ξ = 0.08. The dashed lines are calculated for n̄com = 0.05 from the
propagator (3.29), neglecting pulse shaping and non-resonant carrier excitation. The solid
lines are obtained from numerically solving the Schrödinger equation for time-dependent
Ω(t) and imbalanced Rabi frequencies ξ = 0.08. The parity oscillations for the ground state
cooled ions have an amplitude of 0.990(1) which leads to a fidelity of 0.993(1). The amplitude of the parity oscillations was obtained by a fit with the function P (φ) = A·sin(2φ+φ0 ).
The value of the phase φ0 is without significance. It arises from phase locking the frequencies
ν, ν+ , ν− and could have been adjusted to zero.
61
the possibility of entangling ions irrespective of their motional state by using these gates, there
has not been any experimental demonstration of this gate property up to now. The reason for this
is that independence of the motional state, as predicted by (3.29), is achieved only deep within
the Lamb-Dicke regime whereas experiments demonstrating entangling gates on hyperfine qubits
usually have Lamb-Dicke factors on the order of η = 0.1−0.2 [34, 109, 144]. Therefore, all previous
experimental gate realizations used laser cooling to prepare at least the motional mode mediating
the gate in its ground state with n = 0.
The corresponding time evolution and parity oscillations for ions that are merely Doppler-cooled
to a thermal state are shown in figure 6.4 (c) and (d) respectively. As the coupling strengths on
√
√
the upper and lower motional sidebands scale as ∝ n + 1 and ∝ n, non-resonant sideband
excitation transfers population much faster from |↓↓, ni into |↓↑, n ± 1i, |↑↓, n ± 1i as compared
to the case of ions prepared in the ground state with n̄ = 0. After the gate time tgate =50 µs,
however, the undesired population p1 nearly vanishes as in the case of ground-state cooled ions
and the Bell state Ψ1 is again created. In the experiment, we find a population p1 = 0.012(1) in
the undesired energy eigenstates. The parity oscillations have an amplitude of 0.980(2), resulting
in a Bell state fidelity of 0.984(2). The reasons for the somewhat reduced fidelity as compared
to ground-state cooled ions are only partially understood. In part, the fidelity loss arises from
a variation of the coupling strength on the vibrational sidebands as a function of n caused by
higher-order terms in η. Fitting equations (3.36) to the population evolution data allows us to
determine the mean vibrational quantum number as n̄ = 18(2). This value is consistent with
independent measurements obtained by comparing the time evolution of the ions when exciting
them on the carrier and on the blue motional sideband. For a thermal state with n̄ = 18 and
η = 0.04 calculations show that this effect amounts to additional errors of 5.5 × 10−3 , which makes
up for most of the difference.2
The amplitude of the parity oscillations was obtained by a fit with the function P (φ) = A ·
sin(2φ + φ0 ). The value of the phase φ0 is without significance. It arises from phase locking the
frequencies ν, ν+ , ν− and could have been adjusted to zero by changing the phase of the signal
generators producing ν+ and ν− .
As mentioned in Section 3.5.1, the AC-Stark compensation by unbalancing the power of the
red and blue frequency component is not applicable to ions in a thermal state. Instead, the laser
frequency needs to be adjusted to account for AC-Stark shifts δAC , a technique that works well
as long as the AC-Stark shifts are smaller than the coupling strength λ of the gate interaction
appearing in (3.30) (otherwise, in the case δAC À λ, small laser power fluctuations give rise to
large phase shifts). Therefore, care must be taken to choose the direction and polarization of the
gate laser such that a favorable ratio λ/δAC is obtained. In experiments with a gate duration
of tgate =50 µs on the transition S1/2 , mj = 1/2 → D5/2 , mj = 5/2, we achieved λ/δAC ≈ 3
and needed to shift the laser frequency by about 7.5 kHz for optimal Bell state fidelity. In the
experiments shown in figure 6.4, a further reduction of the AC-Stark shift could be obtained by
using a σ + -polarized laser beam incident on the ions along the direction of the magnetic field. In
this geometry the AC-Stark shift is predominantly caused by the S1/2 ↔ P3/2 dipole transition
loss of fidelity due to coupling strength fluctuations is given by 1 − F = (π/2)2 (δΩ/Ω)2 . The expectation
value for δΩ/Ω can be estimated by assuming a thermal state distribution and a coupling strength that depends
on n in first order as Ω = 1 − η 2 n.
2 The
62
Figure 6.5.: Bell state fidelities after n gate operations applied to the input states |↓↓i(N)
and |↓↑i(•) for tgate = 50µs. Taking into account the error for state preparation of the
input states |↓↑i and a similar error to measure the parity signal, we conclude that the gate
operation works on all tested input states similarly well. The solid lines reflect a Gaussian
decay of the parity fringe amplitudes as a function of the number of gates and a linear decay
in the desired populations caused by the spectral impurity of the laser. For both input states
the gate operation implies errors of less than 0.2 after 21 consecutive applications.
since the D5/2 , mj = +3/2 state does not couple to any of the 4p Zeeman states. Measurements
show that the shift is reduced to about 1.8 kHz without compromising the gate speed.
6.1.4. Different input states and multiple gate operations
For a gate time of 50 µs the analysis was extended by applying the gate to the state |↓↑i which
is prepared by a π/2 rotation (beam 2) of both ions, followed by a π phase-shift pulse on a single
ion performed with the far-detuned focused beam (as explained in chapter 5.4), and another π/2
rotation applied to both ions. This pulse sequence realizes the mapping
|↓↓i −→ | ↓ + ↑i| ↓ + ↑i −→ | ↓ − ↑i| ↓ + ↑i −→ |↓↑i
(6.1)
to the desired input state for the gate. Imperfections of single-ion addressing led to an error in state
preparation of 0.036(3) (Addressing was improved after these experiments had been performed to
the values stated in chapter 5.4). For the Bell state analysis, the population p1 was measured to
exp
infer ρexp
↑↓,↑↓ + ρ↓↑,↓↑ . Unfortunately parity oscillations cannot be introduced by a collective π/2
pulse acting on the state |↑↓i + i |↓↑i. Instead, this state was transferred into |↑↑i + i |↓↓i by
repeating the steps of sequence (6.1) as for the state preparation. The coherence was measured
again by performing parity oscillations.
Figure 6.5 shows a comparison of the fidelity of the gate starting either in |↓↓i or |↓↑i. The
fidelity of a Bell state created by a single gate starting in |↓↑i is 0.95(1). Taking into account the
errors for state preparation and the Bell state analysis we conclude that the entangling operation
works equally well for |↓↑i as an input state. This hypothesis is supported by the observation that
for both states we obtain a similar decay of Bell state fidelities with increasing gate number. The
63
Figure 6.6.: Fidelity as a function of the global frequency detuning of the bichromatic light
pulse from the carrier transition (here, the sideband detuning was set to ²/(2π)=20 kHz). A
maximum fidelity of 0.995(4) was found for a detuning of 500 Hz from the transition center
due to a residual AC-Stark shift. The solid line is obtained by numerically solving equation
(3.27) and taking into account the AC-Stark shift compensation by different powers of the
blue and the red laser frequency component. At the maximum the solid line experiences a
second order frequency dependence of −9.6(3) × 10−9 Hz 2 .
application of N gate operations maps the input state |↓↓i to
|↓↓i −→ |↓↓i + i |↑↑i −→ |↑↑i −→ |↓↓i + i |↑↑i −→ |↑↑i −→ . . .
and the state |↓↑i to
|↓↑i −→ |↓↑i − i |↑↓i −→ |↑↓i −→ |↑↓i − i |↓↑i −→ |↓↑i −→ . . .
Compared with earlier results [40] where multiple gate operations were induced by varying the
duration of a single bichromatic pulse, here up to 21 individual amplitude-shaped pulses were
applied. Splitting up a long pulse into many shorter gate pulses has no detectable effect on the
fidelity of the Bell states produced, and in both cases we obtain a Bell state fidelity larger than
0.80 after 21 gates.
6.1.5. Gate errors
From the preceding measurements the following error sources can be determined and quantified
(see also [40, 76]).
• A bichromatic force with time-dependent Ω(t) acting on ions prepared in an eigenstate of
Sy creates coherent states α(t) following trajectories in phase space that generally do not
close [145, 146]. For the short rise times used in our experiments, this effect can be made
negligibly (< 10−4 ) small by slightly increasing the gate time.
• Spin flips induced by incoherent off-resonant light of the bichromatic laser field reduce the
gate fidelity. The spectrum of the laser determined by a beat measurement (see figure 4.8(b))
64
shows that a fraction γ of about 2·10−7 of the total laser power is contained in a 20 kHz bandwidth B around the carrier transition when the laser is tuned close to a motional sideband. A
simple model predicts spin flips to cause a gate error with probability pf lip = (πγ|²|)/(2η 2 B).
This would correspond to a probability pf lip = 8 · 10−4 whereas the state populations measured for figure 6.5 are consistent with pf lip = 2 · 10−3 . Spin flip errors could be further
reduced by two orders of magnitude by spectrally filtering the laser light and increasing the
trap frequency ω/(2π) to above 2 MHz where noise caused by the laser frequency stabilization
is much reduced.
• Imperfections due to low frequency noise randomly shifting the laser frequency νL with
respect to the atomic transition frequency ν were estimated from Ramsey measurements
on a single ion showing that an average frequency deviation ∆(νL −ν) /(2π) = 160 Hz occurred. From numerical simulations, one can infer that for a single gate operation this
frequency uncertainty gives rise to a fidelity loss of 0.25% (an infidelity of 10−4 would require
∆(νL −ν) /(2π) = 30 Hz). In the parity oscillation experiments shown in figures 6.4 and 6.5,
however, this loss is not directly observable since a small error in the frequency of the bichromatic laser beam carrying out the gate operation is correlated with a similar frequency error
of the carrier ( π2 )φ pulse probing the entanglement produced by the gate so that the phase
φ of the analysing pulse with respect to the qubit state remains well defined.
• Variations in the coupling strength δΩ induced by low-frequency laser intensity noise and thermally occupied radial modes were inferred from an independent measurement by recording
the amplitude decay of carrier oscillations. Assuming a Gaussian decay, a relative variation
of δΩ/Ω = 1.4(1) · 10−2 is found. For m entangling gate operations, the loss of fidelity is
2
2
−4
approximately given by 1 − F = ( πm
to the error of
2 ) (δΩ/Ω) and contributes with 5 · 10
a single gate operation. For the multiple gate operations shown in figure (6.5), this source of
noise explains the Gaussian decay of fidelity whereas laser frequency noise reduces the fringe
amplitude by less than 1% even for 21 gate operations. In combination with error estimates
for state preparation, detection and laser noise, the analysis of multiple gates provides us
with a good understanding of the most important sources of gate infidelity.
• An error that was not investigated in [40] is the dependence of the Bell state fidelity on the
global laser frequency detuning from the qubit transition frequency. Experimental results are
shown in figure 6.6. The solid line fitting the data is calculated by numerically solving the
full Schrödinger equation for different global frequency detunings and evaluating the fidelity.
A second order frequency dependence of −9.6(3)×10−9 Hz 2 is found from calculations at the
maximum point. This suggests that our laser’s typical mean frequency deviation of 160 Hz
contributes with 3 × 10−4 to the error budget.
• A further error source arises when the bichromatic beam couples to both ions with different
strengths. By recording Rabi oscillations simultaneously on the two ions we conclude that
both ions experience the same coupling strength Ω to within 4%. From numerical calculations
we infer an additional error in the measured Bell state fidelity of less then 1 × 10−4 .
• Another possible error source is heating of the COM-mode during the gate operation since
65
the gate is not insensitive to motional heating in the parameter regime of our implementation.
Using the calculation performed in [81], we find a fidelity reduction of ∆F = Γh tgate /2 where
Γh is the heating rate of the COM-mode. As in our experiments Γh = 3s−1 , the fidelity is
reduced by ∆F ≈ 10−4 for tgate =50 µs.
6.1.6. Conclusion
Until recently, entangling gates for optical qubits were exclusively of the Cirac-Zoller type which
require individual addressing of the ions. Compared to this type of gate the Mølmer-Sørensen
gate gives an improvement in fidelity and speed of nearly an order of magnitude. The achieved
fidelity sets a record for creating two-qubit entanglement on demand irrespective of the physical
realization considered so far. The results with concatenations of 21 of these operations bring the
realization of more complex algorithms a step closer to reality. The implementation of a gate
without the need for ground state cooling is of particular interest in view of quantum algorithms
that require entangling gates conditioned on quantum state measurements that do not preserve
the ions’ motional quantum state.
When considering gate imperfections, two regimes are of interest: On the one hand, in view of
a future implementation of fault-tolerant gate operations, it is important to investigate whether
the gate operation allows in principle for gate errors on the order of 10−4 or below. On the other
hand, for current experiments aiming at demonstrating certain aspects of quantum information
processing, errors on the order of 10−2 are not forbiddingly high. For these experiments, the
prospect of carrying out a gate operation using ions that are not in the vibrational ground state
of the mode mediating the internal-state entanglement, is appealing as it might allow to perform
entangling gates after having split a long ion string into shorter segments (the splitting process has
been demonstrated to heat up the ions by no more than a single quantum of motion [42]). Similarly,
quantum state detection by light scattering on a cycling transition heats up the vibrational mode.
However, if done properly, the final mean quantum number stays well below the average of 20
quanta for which we demonstrated entanglement generation. Therefore, experiments involving
gates after splitting and detection operations might profit from a quantum gate as demonstrated
in chapter 6.1.
For future ion trap experiments in the fault-tolerant domain, the needs are going to be different.
Here, ground state laser cooling will most likely be indispensable to achieve the highest fidelity
possible. Also AC-Stark compensation based on imbalanced bichromatic beam intensities should
be avoided as the technique tends to complicate the gate Hamiltonian and to introduce small
additional errors. Even though the current experiments are still limited by technical imperfections,
simulations predict that in principle it should be possible to achieve gate errors of 10−4 or below
with a Mølmer-Sørensen gate on a quadrupole transition. Gates with ions in motional states are
important in this context as no experiment will cool the ions to the ground state n = 0 perfectly
(in current experiments, the ground state is typically occupied with a success rate of 90 to 99%).
Our simulations indicate that for ions in n = 1, gate errors could still be as small as 2 · 10−4 so
that gate errors of 10−4 or below seem feasible even without perfect initialization of the motional
mode.
The optical qubit as used here is certainly not the best solution for long time storage of quantum
66
6.2 Three ion entangled states and arbitrary operations on three qubits
Figure 6.7.: (a) Evolution of the populations off three ions p0 (O), p1 (•), p2 (M) and p3 (¨)
induced by a Mølmer-Sørensen interaction of pulse length τ . Pulse shaping is used to
suppress off-resonant excitations. At 100 µs the maximally entangled state |↓↓↓i − |↓↑↑i −
|↑↓↑i − |↑↑↓i is created. The solid lines are calculated from the propagator (3.29). (b)
Populations p0 (O), p1 (•), p2 (M) and p3 (¨) after a single gate operation (tgate =100 µs) where
the global frequency detuning of the bichromatic entangling pulse is varied by scanning
AO 2. A maximally entangled state is achieved at a detuning of 700 Hz relative to the qubit
transition frequency due to uncompensated AC-Stark shifts. The solid lines are calculated
by solving the Schrödinger equation for the Hamiltonian given in (3.27) amended by a term
accounting for the measured AC-Stark shift.
information. Instead, qubits encoded in two hyperfine ground states whose frequency difference is
insensitive to changes in magnetic field are preferable. These magnetic-field insensitive hyperfine
qubits can store quantum information for times exceeding the duration of the gate operation
presented here by more than four orders of magnitude [102, 147, 148]. However, on such qubit
states no high-fidelity universal gates have been demonstrated so far. By mapping between the
hyperfine qubit encoded in the ion’s ground states and the optical qubit we will benefit from both
of their advantages (see chapter 6.3).
6.2. Three ion entangled states and arbitrary operations on
three qubits
After entangling two ions with a MS interaction the next obvious step is to entangle more ions.
Unfortunately in the current setup we are not able to reliably trap more than three ions in a
crystal-like structure with reasonable trapping parameters. The issue is not yet clear, but most
likely this is due to a high collision rate of the ions with background gas particles. An indication for
this is, that on average every 4 h an ion in a crystal becomes dark due to a chemical reaction with
the background gas. Measurements of the axial oscillation frequency of a two ion crystal with one
dark ion indicated that an OH group attached itself to a dark ion. As we are not able to trap more
than four or more ions the experiments presented in this chapter only deal with entangling three
ions. The results are very similar to the ones recently published by Chris Monroe’s group [149].
The difference is that in this thesis the qubit is an optical one and a single axial oscillator mode is
used for mediating the coupling.
67
Figure 6.8.: (a) A ( π2 )(−y) pulse applied to all ions maps the three ion
¡ π ¢ entangled state
|↓↓↓i − |↓↑↑i − |↑↓↑i − |↑↑↓i onto the GHZ state |↓↓↓i + i |↑↑↑i. A second 2 φ analysis pulse
again applied to all ions gives rise to parity oscillations P (φ) = A · sin(3φ) as a function of φ.
The obvious difference to parity flops with two ions is the different period, three oscillations
occurring for φ changing from 0 to 2 π. A fit with the function P (φ) = A · sin(3φ + φ0 )
yields the parity fringe amplitude A = 0.977(2). The value of the phase φ0 is without
significance. Together with the independently measured populations p1 + p2 = 0.0189(1)
this leads to a GHZ state fidelity of 0.979(2). (b) Parity like oscillations of the states
(|↑↑i1,2 + i |↓↓i1,2 ) ⊗ |↓i3 (◦) and (|↑↓i1,2 + |↓↑i1,2 ) ⊗ |↓i3 (♦) created by entangling two out
¡ ¢
of three ions. The analysis π2 φ pulse was applied to two ions and the state evolution was
analyzed with the camera. The addressing of all operations on two ions is clearly visible
in the states |↑↑i1,2 ⊗ |↑i3 , i |↓↓i1,2 ⊗ |↑i3 (combined in •) and |↑↓i1,2 ⊗ |↑i3 , |↓↑i1,2 ⊗ |↑i3
(combined in N) as they show no oscillatory behavior and are not populated. From these
data one can infer a state fidelity of 0.958(2) for entangling two ions and leaving the third
one untouched.
6.2.1. A three ion Mølmer-Sørensen gate
Three 40 Ca+ ions were loaded into the linear trap. After Doppler cooling and frequency-resolved
optical pumping, the axial center of mass mode is cooled close to the motional ground state.
The ions are now initialized to |↓↓↓i. Then, the gate operation is performed with beam 1 (see
figure 5.4)(a) and the probabilities pk of finding k ions in the state |↓i are measured for a varying
pulse duration. A Rabi frequency of Ω/(2π) ≈ 70 kHz is required for performing a gate operation
with ²/(2π) = 10 kHz and a three-ion Lamb Dicke parameter of η = 0.036. To make the bichromatic
laser pulses independent of the phase ζ, the pulse is switched on and off by using pulse slopes of
2 µs duration.
The time evolution of the three-ion state populations pk are shown in figure 6.7 (a). After
a gate time of tgate =100 µs the undesired populations p0 and p2 nearly vanish and the state
|↓↓↓i − |↓↑↑i − |↑↓↑i − |↑↑↓i is created. In the experiment, we find a population p0 + p2 = 0.0189(1)
in the undesired energy eigenstates. Contrary to the two ion case, an application of two times the
gate does not flip the state of all qubits but rather returns all populations to the initial state.
The AC-Stark shift caused by the bichromatic light is not compensated. A scan of the global
laser frequency reveals a vanishing of the undesired populations p0 + p2 at 700 Hz detuning relative
to the carrier transition. The resulting frequency dependence of the populations after a gate
operation are depicted in figure 6.7 (b).
68
6.2 Three ion entangled states and arbitrary operations on three qubits
For the state Ψ1 = |↓↓↓i − |↓↑↑i − |↑↓↑i − |↑↑↓i, the fidelity is not directly observable with parity
oscillations. First a Ux ( π2 ) pulse is applied to all ions mapping the entangled state |↓↓↓i − |↓↑↑i −
|↑↓↑i − |↑↑↓i onto the GHZ state |↓↓↓i − i |↑↑↑i. A second U ( π2 , φ) analysis pulse again applied to
all ions gives rise to parity oscillations P (φ) = A·sin(3φ) as a function of φ (see figure 6.8 (a)). The
obvious difference to parity flops with two ions is the different period, three oscillations occurring
for φ changing from 0 to 2 π. A fit with the function P (φ) = A · sin(3φ + φ0 ) yields the parity
fringe amplitude A = 0.977(2). Combining this measurement with the previously measured state
populations a fidelity of 0.979(2) is obtained for Ψ1 .
6.2.2. Entangling two out of three ions
The goal for all QIP experiments is to implement arbitrary quantum circuits. Here a first step towards that direction is presented by combining a global MS-interaction with addressed σz rotations
and global carrier pulses as described in chapter 5.4.
The most important operation is to implement the pulse sequence shown in figure 5.4(c). This
pulses sequence effectively switches off the interaction between one ion and the other two. After
the application of these pulses we end up in the state Ψ2 = (|↑↑i1,2 + i |↓↓i1,2 ) ⊗ |↓i3 where the
¡ ¢
subscript denotes the ion. To analyze the fidelity of this state we again apply a π2 φ pulse to
¡ ¢
the two entangled ions by using the same principle. All ions are first rotated by π4 φ , then ion
¡π¢
1 experiences a π phase shift by an AC-stark pulse. The second 4 φ pulse rotates ion 2 and 3
by the desired angle and returns ion 1 to the initial state. This pulse sequence induces parity-like
oscillations between the states Ψ2 and (|↑↓i1,2 + |↓↑i1,2 ) ⊗ |↓i3 . The state of the ions was detected
with the help of the camera and the recorded oscillations are shown in figure 6.8. The addressing of
all operations on two ions is clearly visible in the evolution of the states |↑↑i1,2 ⊗ |↑i3 , |↓↓i1,2 ⊗ |↑i3
and |↑↓i1,2 ⊗ |↑i3 , |↓↑i1,2 ⊗ |↑i3 . These states are not populated, thus they effectively did not
participate in any of the operations. From these data combined with the independently measured
population one can infer a state fidelity of 0.958(2) for the state Ψ2 .
Realizing this pulse sequence opens up the possibility for more complicated sequences as all
the necessary building blocks have been demonstrated. Recent experiments published by the
group of David Wineland demonstrated a different approach to implement arbitrary quantum
circuits [55, 56]. They “designed“ their interactions by moving the ions in a segmented trap and
re-cooling them with a different ion species.
One particularly interesting algorithm is the one for correcting phase errors on a single qubit as
shown in figure 6.9. The first MS gate encodes qubit 1 in a three ion state that is protected against
phase errors. After the error occurred, the following pulses detect and correct for the error. After
the operation only ion 1 is in the corrected state, ion 2 and 3 are in some arbitrary state. Ion
1 can be again protected by resetting ion 2 and 3 to |↓↓i and applying another entangling pulse.
The phase errors that can be corrected are either a phase flip on one ion or small phase errors on
all of them, e.g. a π/10 phase error on all of them is corrected in more than 99.8% of the cases
assuming perfect operations. In this sequence one can identify a building block very similar to the
one entangling two out of three ions. Meanwhile experiments have been performed implementing
this algorithm and will be published soon. Similar algorithms using the same NMR-like techniques
and building blocks were investigated by Volckmar Nebendahl in his thesis [150]. The sequence for
69
Figure 6.9.: Error correction sequence for three ions. The first MS gate encodes qubit 1
in a three ion state. After this encoding a phase error can occur and the subsequent pulses
detect and correct for it. After the operation only ion 1 is in the corrected state, ion 2 and
3 are in some arbitrary state. Ion 1 can be again protected against phase errors by resetting
ion 2 and 3 to |↓↓i and applying another entangling pulse. The phase errors that can be
corrected are either a phase flip on one ion or smaller phase errors on all of them. The grey
box corresponds to the block already implemented, entangling two out of three ions.
the error correction code shown here and multi qubit gates like a Toffoli gate [151] can be found
in his thesis.
6.3. High fidelity entanglement of
43
Ca+ hyperfine clock states
In experiments with trapped ions most of the elementary building blocks for quantum information
processing have been demonstrated. State initialization, long quantum information storage times,
entangling gates and readout have been realized with high fidelity [40, 109, 147, 148, 152, 153].
A major challenge in current experiments is to integrate these building blocks into a single setup
and make them work for a given ion species and parameter range set by the trap frequencies, the
magnetic field strength and further parameters. Quantum information encoded in ground state
atomic levels whose energy difference only weakly depends on changes of the magnetic field (“clock“
states) has been stored for more than a second [147, 148, 154, 155].
Most high fidelity entangling operations have been demonstrated on qubits with limited coherence times [102, 109]. One exception is the entanglement of hyperfine clock states in a Yb+ system
demonstrated by Kim et al. [149].
An attractive way of combining high fidelity entangling gates with long storage times is to
coherently map the quantum information from clock states to atomic states which are suitable for
performing the entangling operation with high fidelity, and to map the information back at the
end of the entangling operation [147]. In case this mapping operation is addressed at individual
ions - for instance by using a strongly focused laser - the entanglement operation can be applied
to the entire ion string.
The coherence time of the optical qubit used for entangling the ions is limited by magnetic
field fluctuations, laser linewidth and ultimately the 1.2 s lifetime of the D5/2 state. For the most
abundant isotope 40 Ca+ , encoding the qubit in long-lived hyperfine clock states is impossible as this
isotope has a nuclear spin I = 0. In the isotope 43 Ca+ on the other hand where I = 7/2, coherence
times of many seconds have been measured for qubits encoded in the S1/2 (F = 4, mF = 0) and
S1/2 (F = 3, mF = 0) states [154, 155].
70
6.3 High fidelity entanglement of
43
Figure 6.10.: (a) Energy levels of 43 Ca+ showing the hyperfine splitting of the atomic
states S1/2 and D5/2 . Microwave radiation applied to an electrode close to the ions drives the
hyperfine qubit encoded in the states |↓i and |↑i. A laser at 729 nm excites the ions on the
transition from the S1/2 (F = 4) to the D5/2 (F = 2, .., 6)-states. It is used for ground state
cooling, state initialization, state discrimination and to excite the optical qubit comprised
of the states |↓i and |⇑i or |⇑’i. (b) Using a bichromatic laser beam, two 43 Ca+ ions are
entangled on an optical transition with a maximum target state fidelity of 96.9(3)% at a
magnetic field of 6 G. By introducing a waiting time between the creation and the analysis
of the Bell state we observe a Gaussian decay of the Bell state fidelity (•) (black solid line).
Data for the first 3 ms are displayed as inset. When the optical qubit is mapped to the
hyperfine qubit subsequent to the gate operation we obtain a maximum Bell state fidelity
of 96.7(3)% and the lifetime of the entanglement increases to 96(3) ms (F). Measurements
taken at a magnetic field of 3.4 G give similar results (N) for the coherence time. Here, the
optical qubit |↓i ↔ D5/2 (F = 4, mF = 2) in combination with the laser beam 3 was used for
entangling the ions. Each data point represents 18.000 to 24.000 individual measurements.
In this chapter a combination of the long quantum information storage times observed for the
hyperfine qubit of a single 43 Ca+ ion with the high fidelity gate operation on optical qubits is
investigated by mapping between these two qubits. The steps for state initialization and Bell
state preparation in the optical qubit are discussed. Then the mapping to the hyperfine qubit is
described and for both qubits the entanglement decay is measured. Remapping to the optical qubit
after a 20 ms storage time and the application of further gate operations is demonstrated. Due
to the nuclear spin of I = 7/2, the isotope 43 Ca+ has a relatively complex energy level structure
compared to other ions used for quantum information processing. Magnetic field and polarization
settings are discussed to reduce errors due to close by Zeeman-levels, which are especially harmful
for gate operation in the case when large coupling strengths are required in a relatively short
time. The findings of this chapter were also published in reference [77]. Similar techniques were
implemented in experiments conducted in David Wineland’s group [55, 56].
6.3.1. Qubit preparation and manipulation
An energy level scheme for 43 Ca+ with all relevant hyperfine levels is shown in figure 6.10 (a). In the
experiments reported here, the hyperfine qubit, comprised of the states |↑i ≡ S1/2 (F = 3, mF = 0)
and |↓i ≡ S1/2 (F = 4, mF = 0), is driven by applying a microwave field (3.2 GHz) to one of
the electrodes which is also used to compensate for external electric stray fields (see figure 4.1
71
electrodes (C)).
Each experimental sequence starts by Doppler-cooling the ions for 3 ms followed by sideband
cooling of both axial modes close to the ground state (n̄com , n̄st ≈ 0.03(3), 0.4(1); 5 ms and 3 ms).
Optical pumping (as described in chapter 5.4) initializes both ions in the energy level S1/2 (F =
4, mF = 4) in more than 98.9% of the cases.
Subsequently the populations of both ions are transferred by a π-pulse to the energy level
D5/2 (F = 3, mF = 2) (transfer 1). By turning on the lasers at 397 nm and 866 nm for 500 µs
one can detect residual population remaining in the S1/2 state manifold (PMT check). All cases
where more than 3 photons are detected are rejected (∼ 20 photons/ms are received per ion in
the S1/2 -state at a dark count rate of 0.2 photons per ms). The rejection rate is typically 2%.
Initialization of the hyperfine qubit is completed by another π-pulse (transfer 2) on the transition
D5/2 (F = 3, mF = 2) ↔ S1/2 (F = 4, mF = 0) ≡ |↓i. In total the initialization of internal and
external degrees of freedom takes about 12 ms and the qubit state |↓↓i is populated with a fidelity
of more than 99.2%. The fidelity of state initialization to |↓↓i is measured by transferring the
population with two consecutive π-pulses from |↓↓i to two different Zeeman-levels in the D5/2
manifold (D5/2 (F = 3, mF = 2) and D5/2 (F = 5, mF = 2)) and determining the remaining
population in the S1/2 by fluorescence detection. The fidelity of optical pumping is determined in
a similar way.
The magnetic field is set to 6.0 G in order to achieve a sufficiently large frequency separation of
the neighboring transition lines in the spectrum of the S1/2 ↔ D5/2 quadrupole transition [126].
The transition |↓i ↔ D5/2 (F = 6, mF = 1) ≡ |⇑i is chosen as optical qubit because of its small
magnetic field sensitivity of 350 kHz/G (which is at least a factor of two lower than the sensitivity
of any mF = 0 to mF = 0 transition for the chosen magnetic field) and in order to avoid having
coinciding resonances with (micro)motional sidebands of other spectral components. Moreover,
this choice of optical qubit has the advantage of a comparatively large Clebsch-Gordan coefficient
which together with a carefully set laser polarization (for this qubit we use beam 1 as shown in
figure 5.4(a)) suppresses neighboring Zeeman transitions with ∆m 6= +1. The transitions ∆m = 0
and 2 are suppressed by factors of 38 or more in terms of their Rabi frequencies, the strongest
being the |↓i ↔ D5/2 (F = 6, mF = 0) transition, whose resonance is more than 4 MHz away from
the |↓i ↔ |⇑i transition. The two closest transitions are S1/2 (F = 4, mF = ±1) ↔ |⇑i (& 2 MHz
away) which are suppressed by a factor of 270 in coupling strength. The high coupling strength
on the gate transition reduces the required power for the gate operation and thus the AC-Stark
shift caused by off-resonant coupling to dipole transitions.
After initialization to |↓↓i, a Mølmer-Sørensen entangling operation (MS 1) consisting of a
single bichromatic laser pulse is applied to the optical qubit transition to create a Bell state of
|↓↓i + i |⇑⇑i. The gate time τgate = 100 µs corresponds to a detuning of 10 kHz from the axial
sideband. For the chosen beam geometry, these settings lead to an AC-Stark shift of 3.5 kHz which
is fully compensated by using different coupling strengths Ωb , Ωr for the red and the blue detuned
frequency component with ξ = 0.14.
72
6.3 High fidelity entanglement of
43
Table 6.1.: Duration and errors of each experimental step and the achieved state fidelities.
Experiment
Duration Error
State
(µs)
(%)
fidelity (%)
step
Opt. pumping
600
< 1.1
> 98.9
Transfer 1
20
0.7
98.2
PMT check
500
0.7
99.3
Transfer 2
20
< 0.1
> 99.2
Total prep. |↓i
1140
< 0.8
> 99.2
MS 1
100
2.3
96.9
Map
120
0.2
96.7
Wait
20 × 103
2.2
94.6
Map−1
120
0.7
93.9
MS 2
100
3.7
90.4
MS 3
100
2.9
87.8
6.3.2. Experimental results
The fidelity of the created Bell state is determined by measuring the populations p0 +p2 in |↓↓i and
|⇑⇑i and the coherence between the two states. The coherence is again inferred from the amplitude
of parity oscillations. The maximum fidelity was determined to 96.9(3)%. The durations and errors
of all experimental steps and the achieved state fidelities are given in table 6.1.
By introducing a waiting time before the Bell state analysis, a Gaussian decay of the parity
fringe contrast is observed. From this model a Bell state fidelity of 75% after 3.43(5) ms is
extrapolated (see inset of figure 6.10(b)) corresponding to a loss of 50% of the coherence since the
state populations hardly change. The magnetic field sensitivity of this optical qubit is 350 kHz/G
which is one cause of dephasing. However, in measurements where a single 43 Ca+ ion is probed on
the first order field insensitive transition (S1/2 (F = 4, mF = 4) ↔ D5/2 (F = 4, mF = 3) at 3.4 G)
a single qubit coherence times of 8.1(3) ms is obtained (see figure 4.8(a) inset). This leads to the
conclusion that the decoherence on the optical qubit is also caused by acoustical noise picked up
by the 2 m fiber cord used and the finite linewidth (∼ 20 Hz) of the laser.
Mapping the optical qubit to the hyperfine qubit is achieved by a microwave π-pulse on the hyperfine qubit transition followed by a π-pulse on the optical qubit (map) which results in an average
error rate of 0.2%. Detection of the hyperfine qubit states is done by shelving the population of |↓i
to the D5/2 manifold by two consecutive π-pulses to different Zeeman states (D5/2 (F = 5, mF = 2)
and D5/2 (F = 6, mF = −2)) followed by fluorescence detection. A Bell state fidelity measurement
after the mapping yields a parity fringe contrast of 95.7% - induced by varying the phase of a
microwave π/2-pulse - corresponding to a target state fidelity of still 96.7%. By delaying the state
analysis a Gaussian decay of the parity fringe pattern is observed (figure 6.10) corresponding to a
50% loss of phase coherence on a time scale of 96(3) ms which means that a factor 28 in coherence
time is gained compared to the optical qubit.
For a successive gate application the Bell state is mapped back to the optical qubit (map−1 )
after a waiting time of 20 ms and the gate (MS 2) is applied a second time, disentangling the ions
to |⇑⇑i in 90.4% of the cases, indicating an error of 3.7%. Another subsequent application of the
gate (MS 3) adds an error of 2.9% leading to |↓↓i − i |⇑⇑i with a fidelity of 87.8%.
The results with successive gates suggest that the Bell state fidelity depends slightly on whether
73
the gate is applied to the input state |↓↓i or |⇑⇑i, an effect already observed in previous experiments.
This observation can be explained by the different spectator states into which the population
can leak by unwanted excitations. The effect is most prominent in a series of measurements
where the magnetic field was set to 3.4 G. For this field strength the Zeeman splitting of the
ground state and the axial trap frequency coincide to within 10 kHz. Using the transition |↓i ↔
D5/2 (F = 4, mF = 2) ≡ |⇑’i as optical qubit the polarization of beam 3 (see figure 5.4(a)) is
adjusted to suppress the coupling strength to the nearest neighboring transitions. Note that in
this geometry and polarization setting, both ∆m = ±2 have optimal coupling, so a higher gate
laser power is required as compared to |⇑i. This leads to Bell state fidelities of up to 96.3% after
a single gate operation applied to |↓↓i . The decoherence for these Bell states after mapping to
the hyperfine qubit is also shown in figure 6.10(b). Applying the gate operation on the input
state |⇑’⇑’i the maximum obtained Bell state fidelity after a single gate operation is only 92%.
Further measurements at a magnetic field of 1.36 G where we expected no coincidences of spectral
components reveal a maximum Bell state fidelity of 95%. For these measurements, beam 1 was
used again with |↓i ↔ |⇑i as the optical qubit.
6.4. Comparison between
40
Ca+ and
43
Ca+ .
Several experiments including this thesis have demonstrated that 40 Ca+ and 43 Ca+ ions are suited
for QIP processing. Most DiVincenzo criteria including high fidelity entangling gates have been
demonstrated for both isotopes. 40 Ca+ has the advantage of a very simple level structure but lacks
the possibility to encode qubits in states with low magnetic field dependence. Furthermore the
coherence time of the optical qubit suffers from laser frequency noise. Handling 43 Ca+ is much
more difficult due to its complicated level structure. The advantage of 43 Ca+ is that qubits can be
encoded in hyperfine and optical states with low magnetic field dependence. The hyperfine qubit
has an additional advantage as it can be manipulated by either a Raman laser or a microwave field
such that the frequency instability of the driving field is negligible.
Comparing the results for entangling 43 Ca+ ion with the results obtained for entangling 40 Ca+
ions, one can conclude that low errors of the Mølmer-Sørensen interaction on the optical qubit
require that the qubit transition is well isolated from other spectral components such that no
other transitions are erroneously excited. This view is supported by the observation that the gate
mechanism favors a high magnetic field, where the transitions to neighboring lines are spectrally
well separated. However, the sensitivity to magnetic field fluctuations of the hyperfine qubit
increases for higher fields. Therefore, a particularly interesting regime for 43 Ca+ ions occurs at a
magnetic field of 150 G where nonlinearities in ground state Zeeman splitting lead to a first order
magnetic field insensitive transition S1/2 (F = 4, mF = 0) ↔ S1/2 (F = 3, mF = 1). This promises
even longer coherence times as well as smaller errors for all spectrum dependent operations since
the Zeeman splitting is huge compared to the measurements presented here.
The following points summarize and compare the operations used to handle the optical qubit in
40
Ca+ and the optical and hyperfine qubit in 43 Ca+ . Additionally the coherence times achieved
in this experiment for different qubits are given.
74
6.4 Comparison between
40
Ca+ and
Doppler-cooling is more complicated for
•
40
•
43
43
43
Ca+ .
Ca+ :
Ca+ : One π-polarized 397 nm laser and one 866 nm laser is sufficient for Doppler-cooling.
Ca+ : Multiple frequency components in the σ + -polarized 397 nm laser and the 866 nm
laser are necessary for Doppler cooling. Additionally, a second π-polarized 397 nm laser is
needed to avoid population trapping in dark states.
State initialization of the
43
Ca+ hyperfine qubit is more complicated than initializing the optical
qubit:
• optical qubit: Optical pumping with a σ + -polarized 397 nm is sufficient to initialize the state
of 40 Ca+ to S1/2 , m = 1/2 or the state of 43 Ca+ to S1/2 , F = 4, mF = 4.
• hyperfine qubit: To initialize the state S1/2 , F = 4, mF = 0 two additional π pulses on the
quadrupole transition or four microwave π pulses are necessary.
The Mølmer-Sørensen gate operation works better for
•
40
•
43
40
Ca+ ions:
Ca+ : A Mølmer-Sørensen gate operation has been achieved which generated Bell states
with fidelities exceeding 99%. [40]
Ca+ : The gate fidelity is limited by off-resonant excitation of close-by Zeeman-states to
about 97%. [77]
The coherence properties of 43 Ca+ qubits are better than the coherence properties of 40 Ca+ qubits:
•
40
Ca+ optical qubit: The coherence time for a single ion is limited to about 3 ms by magnetic
field fluctuations and laser frequency noise.
•
43
•
43
Ca+ optical qubit: Due to the availability of magnetic field independent transitions the
coherence time is only limited by the laser stability. For a single ion a coherence time of 8 ms
was measured.
Ca+ hyperfine qubit: For the hyperfine qubit S1/2 , F = 4, mF = 0 ↔ S1/2 , F = 3, mF = 0
a coherence time of several seconds has been measured. [154]
Which isotope is chosen for a particular experiment depends on what sets the experimental limit
in the sequences that should be run. If the sequence is short and one is interested in the best
overall fidelity (considering the current experimental status) and the easiest handling, then 40 Ca+
should be used. If the limit is set by the coherence time then one should consider using 43 Ca+ . It
is not absolutely necessary to use the hyperfine qubit in 43 Ca+ . The coherence time can already
be increased by choosing a magnetic field independent transition on the optical qubit in 43 Ca+ .
This has the advantage that the handling is still quite simple and one avoids the complicated read
out and initialization steps of the hyperfine qubit.
75
76
7. Experimental test of quantum
contextuality
Since the beginning of quantum mechanics (QM) it has been debated whether hidden variable
(HV) theories can account for the intriguing features of QM [64, 156]. Especially the indeterminism of QM was a feature some physicists could not accept and was one of the reasons why HV
theories came up. In the sixties, Bell found that local HV theories cannot reproduce the quantum
mechanical correlations for local measurements on entangled states [65]. Later, a series of experiments confirmed a conflict between HV and QM theories by realizing a refined version of Bell’s
“Gedankenexperiment“ [66–69]. These experimental refutations led to additional assumptions and
restrictions for the structure of local HV theories.
Kochen, Specker and Bell [70–72] proposed another HV theorem testing for non-contextuality of
measurements. Non-contextuality is the property of a HV model that the value of a measurement
v(A) is determined, regardless of which other compatible measurement is measured jointly with
A. Two or more measurements (A,B,C,. . .) are compatible if they can be measured jointly on the
same individual system without disturbing each other. Compatible measurements can be made
simultaneously or in any order and can be repeated any number of times on the same individual
system and must always give the same result independently of the initial state of the system.
A context is a set of compatible measurements. A physical model is called non contextual if it
assigns a measurement result independent of which other compatible measurements are carried out
on the probed system. Non contextuality is a plausible assumption for all physical models especial
for measurements on distant systems or in the case of measurements concerning different degrees
of freedom on the same system. The KS theorem states that noncontextual HV theories cannot
reproduce the predictions of QM.
For some time it has been debated whether the Kochen Specker (KS) theorem could be experimentally tested at all [157, 158]. Nevertheless an experiment has been proposed testing for
these HV theories [159]. Here the first experiment testing a complete Kochen Specker theorem [75]
falsifying non-contextual HV theories is presented. Recently additional results with photons [73]
and NMR qubits [74] have shown similar results.
In the following chapter the notions compatibility and contextuality as well as the KS theorem
will be discussed. Then the experimental realization of quantum non-demolition measurements
and a test of the Kochen Specker theorem will be shown. At the end of this chapter the results will
be analyzed and possible loopholes will be discussed. The first section is taken from reference [160]
whereas the experimental findings were already published in reference [75].
77
Experimental test of quantum contextuality
7.1. The Kochen-Specker theorem
7.1.1. Measurement scenario
The situation in the experiment will be the following: On an individual system a sequence of
dichotomic measurements (outcomes ± 1) will be performed. The question that arises is: under
which conditions can the result of such measurements be explained by HV theories? Or more
precisely, which conditions a HV model has to violate to reproduce the QM predictions? The HV
model will be described by a distribution p(λ) with the hidden variable λ ∈ Λ from a set Λ. The
distribution summarizes all information about the past, including all preparation steps and all
measurements already performed. Causality is assumed, so the distribution is independent of any
event in the future. It rather determines all the probabilities of the results of all possible future
sequences of measurements. We assume that, for a fixed value of the HV, the outcomes of future
sequences of measurements are deterministic, hence all indeterministic behavior stems from the
probability distribution.
7.1.2. Notation
In an experiment the state of the system is prepared in a repeatable way. In the language of HVmodels this leads to a probability distribution pexp (λ) that can be repeatedly produced even though
its actual form is unknown. In a single instance of an experiment, one obtains a state determined
by the HV λexp . Which λexp has been prepared cannot be controlled by the experimentalist and
the HV is distributed according to pexp (λ).
The following notation will be used to describe measurements and their outcomes. Ai denotes
the measurement of the property A at the position i in the sequence. A sequence of measurements
is denoted by A1 B2 C3 when A is measured first, then B and finally C. The result of a specific
measurement e.g. B2 in such a sequence is written as v(B2 |A1 B2 C3 ). In addition v(A1 B2 C3 ) =
v(A1 |A1 B2 C3 )v(B2 |A1 B2 C3 )v(C3 |A1 B2 C3 ). For the probability distribution p(λ), one can define
probabilities like p(B2+ |A1 B2 C3 ) [or p(B2+ C3− |A1 B2 C3 )] denoting the probability of obtaining the
value B2 = +1 [and C3 = −1] when the sequence A1 B2 C3 is measured. Then, one can also
consider mean values like hB2 |A1 B2 C3 i = p(B2+ |A1 B2 C3 ) − p(B2− |A1 B2 C3 ), or the mean value of
the complete sequence, hA1 B2 C3 i = p[v(A1 B2 C3 ) = +1] − p[v(A1 B2 C3 ) = −1].
7.1.3. Compatibility of measurements
First the notion of compatibility has to be defined. In the simplest case, this is a relation between
a pair of observables, A and B. For that, let SAB denote the (infinite) set of all sequences, which
use only measurements of A and B, that is, SAB = A1 , B1 , A1 A2 , A1 B2 , B1 A2 , . . .. Then one can
formulate:
Definition 1. Two observables A and B are compatible, denoted by A ∼ B, if the following two
conditions are fulfilled:
(i) For any instance of a state (that is, for any λexp ) and for any sequence S ∈ SAB the obtained
78
7.1 The Kochen-Specker theorem
values of A and B remain the same, i.e.
v(Ak |S) = v(Al |S),
v(Bm |S) = v(Bn |S),
where k,l,m,n are all possible indices for which the considered observable is measured at the
positions k,l,m,n in the sequence S.
(ii) For any state preparation (that is for any pexp (λ)), the mean values of A and B during the
measurement of any two sequences S1 , S2 ∈ SAB stay the same, i.e.
hAk |S1 i = hAl |S2 i ,
hBm |S1 i = hBn |S2 i .
Conditions (i) and (ii) are necessary conditions for compatible observables. Two observables which
violate one of them cannot be called compatible.
It is important to note that the compatibility of two observables is experimentally testable by
preparing all possible λexp or pexp (λ). The fact that these sets or the set SAB may be infinite is
not a specific problem here, as any measurement device or physical law can only be tested in a
finite number of cases.
One should note, that (ii) does not necessarily follow from (i), as will be illustrated by the
following example: Consider a HV model where, for any λ, all v(Ak |S) are +1 when the first
measurement in S is A1 , while they are −1 when the first measurement is B1 . The values v(Bm |S)
are always +1. Then, condition (i) is fulfilled, while (ii) is violated, since hAi = 1 but hA2 |B1 A2 i =
−1.
Finally, it should be added that the notion of compatibility can be extended to three or more
observables. For instance, if three observables A,B,C are investigated, one considers the set SABC
of all measurement sequences involving measurement of A,B or C and extends conditions (i) and
(ii) in an obvious way.
7.1.4. Operational definition of noncontextuality
Noncontextuality means that the value of any observable A does not depend on which other
compatible observables are measured jointly with A. For the model, noncontextuality is formulated
as a condition on a HV model as follows:
Definition 2. Let A,B,C be observables in a HV model, where A is compatible with B, and A is
also compatible with C: The HV model is noncontextual if it assigns, for any λ an outcome of A
which is independent of whether B or C is measured before or after A, that is,
v(A1 |A1 B2 ) = v(A2 |B1 A2 ) = v(A1 |A1 C2 ) = v(A2 |C1 A2 ).
(7.1)
Hence, for these sequences one can write down v(A) as being independent of the sequence. If the
condition is not fulfilled, the model is called contextual. If A is an element of several larger sets of
compatible observables, the conditions in equation (7.1) can be extended in a straightforward way.
79
Note that the most important assumptions in (7.1) are that v(A1 |A1 B2 ) = v(A2 |B1 A2 ) and
v(A1 |A1 C2 ) = v(A2 |C1 A2 ). If these are fulfilled, the remaining equality follows from the physically
very plausible assumption, that the result of A1 should not depend on which other observable is
measured after A1 (this observable may be compatible with A or not). It is important to note
that the condition (7.1) is an assumption about the model and — contrary to the definition of
compatibility — not experimentally testable. This is due to the fact that for a given instance of a
state (corresponding to some unknown λ) the experimenter has to decide whether to measure A
or B first.
In the definition, condition (7.1) is assumed for any λ. One may wonder whether this requirement
is too strong, since in any experiment only certain λexp can be tested. Assuming, however, that
there are certain λ which can never occur as a λexp does not make much sense, as in such a HV
theory these λ cannot affect any experimentally observed measurement result, as any pexp (λ) has
to assign a vanishing probability density to them. Hence, one can disregard this case. Finally,
let us add here that if larger sets of compatible observables are considered, definitions (1) and
(2) have to be extended to longer sequences. For instance, if the observables in each of the sets
{A, B, C} and {A, a, α} are all compatible, then, the condition that the assignment of a value to
A is independent of the context results in conditions similar to equation (7.1), but with sequences
of length three.
v(A1 |A1 B2 C3 ) = v(A2 |B1 A2 C3 ) = v(A1 |A1 a2 α3 ) = v(A2 |α1 A2 a3 ) = . . .
(7.2)
7.1.5. The Kochen Specker inequality for noncontextual models
The KS theorem states that noncontextual HV models cannot reproduce all predictions of QM.
Here one (out of many) possible inequalities is discussed involving compatible measurements, which
holds for the QM case but is violated by noncontextual HV theories.
Let’s consider the following array (Mermin Peres square [161, 162]) of observables of a four-level
quantum system (for instance two spin- 12 -particles)
(1)
A = σz ⊗ I (2)
(2)
a = I (1) ⊗ σx
(2)
(1)
α = σz ⊗ σx
(2)
B = I (1) ⊗ σz
(1)
b = σx ⊗ I (2)
(1)
(2)
β = σx ⊗ σz
(1)
(2)
C = σz ⊗ σz
(1)
(2)
c = σx ⊗ σx
(2)
(1)
γ = σy ⊗ σy
(7.3)
and the measurement products of each row Rk and column Ck given by
R1 = v(A) · v(B) · v(C)
R2 = v(a) · v(b) · v(c)
C1 = v(A) · v(a) · v(α)
C2 = v(B) · v(b) · v(β)
R3 = v(α) · v(β) · v(γ)
C3 = v(C) · v(c) · v(γ).
(k)
Here, σi
(7.4)
denotes the Pauli matrix acting on the k-th particle, and all the observables have the
outcomes ±1. Moreover, in each of the rows or columns of (7.3), the observables are compatible.
In any row or column, their measurement product Rk or Ck equals 1, except for the third column
Q
where it equals −1. Therefore, quantum mechanics yields for the product k=1,2,3 Rk Ck a value
of −1, in contrast to non-contextual models.
80
7.2 Experimental realization of QND measurements
Figure 7.1.: Experimental measurement scheme. a For the measurement of the jth row
(column) of the Mermin-Peres square (7.3), a quantum state is prepared on which three
consecutive QND measurements Mk , k = 1, 2, 3, are performed measuring the observables
Ajk (Akj ). Each measurement consists of a composite unitary operation Uk that maps
(1)
(2)
the observable of interest onto one of the single-qubit observables σz or σz which are
measured by fluorescence detection. The unitary operations U are synthesized from singlequbit and maximally entangling gates. In the lower line, σi , σ̃i symbolize, respectively, the
Hamiltonian acting on the qubit and on the D5/2 Zeeman subspace spanned by {| ↓i, |ai} =
{S1/2 , mj = 1/2, D5/2 , mj = 5/2} which is used for hiding one ion’s quantum state from
fluorescence light during detection[49].
To test this property, it has to be expressed as an inequality since no experiment yields ideal
quantum measurements. Recently, it has been shown that the inequality
hXKS i = hR1 i + hR2 i + hR3 i + hC1 i + hC2 i − hC3 i ≤ 4
(7.5)
holds for all non-contextual theories [159], where h· · · i denotes the ensemble average. Quantum
mechanics predicts for any state that hXKS i = 6, thereby violating inequality (7.5). For an experimental test, an ensemble of quantum states Ψ needs to be prepared and each realization subjected
to the measurement of one of the possible sets of compatible observables. Here, it is of utmost
importance that all measurements of Aij are context-independent [159], i.e., Aij must be detected
with a quantum non-demolition (QND) measurement that provides no information whatsoever
about any other compatible observable.
7.2. Experimental realization of QND measurements
As pointed out in the preceding theory section QND measurements are necessary to test the Kochen
Specker theorem. The two qubit observables listed in table 7.3 can be measured in a non destructive
way by applying a unitary transformation U to the state Ψ prior to measuring σz on one ion, and
its inverse operation U † after the measurement (see chapter 2.1.3). The gate decompositions of
the unitary operations necessary to perform all two qubit observables for the Mermin Peres square
are given in table 7.1. These pulse sequences will be used in the chapter 7.3 to determine all
observables for a test of contextual hidden variable theories. The decompositions of Uij into the
elementary gate operations available in our setup were found by a gradient-ascent based numerical
search routine [135]. The unitary transformations are designed such that the information about
the observable is mapped onto a single ion (in this case ion 1). The state detection of ion 1 only
ensures that a single bit of information is acquired. The final inverse operation guarantees that
81
U [σx ⊗ σx ] = UyM S (− π2 )Uz1 ( π2 )
U [σx ⊗ σz ] = UxM S (− π2 )Ux ( π2 )
U [σy ⊗ σy ] = UxM S (− π2 )Uz1 ( π2 )
U [σz ⊗ σx ] = UyM S (− π2 )Ux (− π2 )
U [σz ⊗ σz ] = UyM S (− π2 )Uz1 ( π2 )Uy ( π2 )
Table 7.1.: Gate decomposition of the mapping operations U [σi ⊗ σj ] employed for measuring the five two-qubit spin correlations in the Mermin Peres square.
the two ions are in an eigenstate of the measurement.
It is important to note that some of the observables listed in table 7.3 require unitary transformations that are two qubit entangling gates (see table 7.1). As it turns out the most complicated
series of measurements (row 3 and column 3 in the Mermin Peres square) requires a total of 6
two-qubit gate operations. This, by it self, is already quite demanding but gets even more difficult
by the fact that the state of one ion has to be detected in between the unitary transformations.
The problem that arises with state detection is that the ions do not remain in the ground state but
get heated to the Doppler limit. Fortunately the Mølmer-Sørensen gate operation is insensitive to
the motional state of the ions (see chapter 6.1). This enabled us to omit complicated re-cooling
schemes, as demonstrated in [55], in the experimental realization of the QND measurements.
The measurement scheme that has to be employed to measure one row or column of the Mermin
Peres square can be seen in figure 7.1. In the following this scheme will be explained in more detail
by an example where (|↓i − |↑i)1 ⊗ (|↓i − |↑i)2 is created as an input state and the last row in the
Mermin Peres square is measured.
For this example and the experiment described in the following section, a pair of 40 Ca+ ions is
trapped in a linear Paul trap with axial and radial vibrational frequencies of ωax = (2π) 1.465 MHz
and ωr ≈ (2π) 3.4 MHz in a magnetic field of B = 4 Gauss. The ions are Doppler-cooled by
exciting the S1/2 ↔ P1/2 and P1/2 ↔ D3/2 dipole transitions. Optical pumping initializes an ion
with a fidelity of 99.5% to the qubit state | ↓i ≡ |S1/2 , mj = 1/2i, the second qubit state being
| ↑i ≡ |D5/2 , mj = 3/2i.
The unitary operation to create the input state (|↓i − |↑i)1 ⊗ (|↓i − |↑i)2 is a π/2-pulse on
both ions UΨ = Uy ( π2 ). After the state preparation the first QND measurement, determining the
parity σz ⊗ σz of the input state, has to be carried out. This can be done by applying a unitary
transformation which is equivalent to a CNOT operation (see chapter 2.1.3) followed by a PMT
detection of ion 1. This unitary operation mapps the parity of the state on ion 1 and is given in
terms of gate operations available in the experiment: U1 = UyM S (− π2 )Uz1 ( π2 )Uy ( π2 ) which can be
written in matrix representation as

1 0

 0 1
U1 = 
 i 0

0 i
0
−1
0
i

1

0 
.
−i 

0
(7.6)
This operation maps the input state to |↓↓i − i |↑↑i. Detecting the state of ion 1 will result
in equally finding |↓i and |↑i. |↓i will be identified with the +1 eigenstate and |↑i with the -1
82
7.3 Testing the Kochen Specker inequality
eigenstate. By averaging over many realization the measurement shows that the parity of the
input state is zero.
State detection of ion 1 is achieved by fluorescence detection. Ion 2 is hidden from the fluorescence
laser by transferring the population in the state |↓i = S1/2 , mj = 1/2 to the D5/2 , mj = 5/2 state.
The hiding is done by using a pulse sequence similar to the one shown in figure 5.4(b). The only
difference is that the π4 pulses are replaced by π2 pulses such that the pulse sequence reads as
(2)
(1)
Ux (π) = Ux ( π2 )Uz (π)Ux ( π2 ). The same series of pulses is used to un-hide ion 2 after the state
detection. Additional information on the hiding procedure can be found in appendix A.2
The inverse unitary operation to create an eigenstate of the measurement is achieved by applying the pulse sequence U1† = U−y ( π2 )Uz1 ( π2 )UyM S (− π2 )Uz1 (π) which can be written in matrix
representation as


i 0
1 0


 0 i
0 1 
†
.

(7.7)
U1 = 

 0 −i 0 1 
i 0 −1 0
If ion 1 was detected in the state |↓i this operation maps the two ions onto the entangled state
|↑↑i+|↓↓i. Similarly if ion 1 was detected in the state |↑i the two ions are mapped onto the state
|↑↓i+|↓↑i. Subsequently the other two QND measurements are performed in the same way. With
this example one can clearly see that entanglement is created during the measurement process even
though the initial state was a product state.
7.3. Testing the Kochen Specker inequality
√
Equipped with the tools described in chapter 5.4 and 6.1, the singlet state Ψ = |↑↓i − |↓↑i / 2
(1)
MS π
( 2 , 0) to the initial state |↓↓i.
was created by applying the gate-operations Uz (π)U ( π2 , 3π
4 )U
Then the three observables of a row or column of the Mermin-Peres square were measured consecutively (as describe in chapter 7.2). The results obtained for a total of 6,600 copies of Ψ are
visualized in figure 7.2. The three upper panels show the distribution of measurement results
{v(Ai1 ), v(Ai2 ), v(Ai3 )} and their products for the observables appearing in the rows of (7.3), the
three lower panels show the corresponding results for the columns of the square. Subplot f demon(2)
(1)
(2)
(1)
(1)
(2)
strates that Ψ is a common eigenstate of the observables σx ⊗ σx , σy ⊗ σy , σz ⊗ σz , as
only one of the spheres has a considerable size. Five of the correlations have a value close to
+1 whereas hC3 i = −0.91(1). By adding them up and subtracting hC3 i, one finds a value of
hXKS i = 5.46(4) > 4, thus violating equation (7.5).
To test the prediction of a state-independent violation, the experiment was repeated for nine
other quantum states of different purity and entanglement. Figure 7.3 shows that indeed a stateindependent violation of the Kochen-Specker inequality occurs, hXKS i ranging from 5.23(5) to
5.46(4). It was also checked that a violation of (7.5) occurs irrespective of the temporal order of
the measurement triples.
Figure 7.4 shows the results for all possible permutations of the rows and columns of (7.3) based
on 39,600 realizations of the singlet state. When combining the correlation results for the 36
possible permutations of operator orderings in equation (7.3), all values for the Kochen-Specker
83
b
〈R1〉 = 0.92(1)
0.00
c
〈R2〉 = 0.93(1)
-0.01
-0.03
0.02
σx⊗σx
σz ⊗ σ z
d
σ z⊗
σx
Ι
Ι
e
〈C1〉 = 0.90(1)
Ι⊗
-0.03
-0.01
σx
σx
⊗
σz
f
〈C2〉 = 0.89(1)
0.01
σz ⊗ σ x
0.06
⊗
σ z⊗
-0.95
-0.10
σx
〈C3〉 = -0.91(1)
-0.07
σx ⊗ σ z
σz
-0.88
-0.94
-0.92
Ι⊗
-0.01
σy⊗σy
0.00
〈R3〉 = 0.90(1)
-0.92
σ y ⊗ σy
a
-0.89
Ι⊗
σx
Ι
σ z⊗
σx
⊗
σz
Ι⊗
Ι
σx
⊗
σx
σ z⊗
σz
Figure 7.2.: Measurement correlations for the singlet state. a This subplot visualizes the
(2)
(1)
(2)
(1)
consecutive measurement of the three observables A11 = σz , A12 = σz , A13 = σz ⊗ σz
corresponding to row 1 of the Mermin-Peres square. The measurement is carried out on
1,100 preparations of the singlet state. The volume of the spheres on each corner of the cube
represents the relative frequency of finding the measurement outcome {v1 , v2 , v3 }, vi ∈ {±1}.
The exact probabilities for finding {v1 , v2 , v3 } are given in appendix A.1 . The color of the
sphere indicates whether v1 v2 v3 = +1 (green) or −1 (red). The measured expectation values
of the observables A1j are indicated by the intersections of the shaded planes with the axes of
the coordinate system. The average of the measurement product hR1 i is given at the top. b-f
Similarly, the other five subplots represent measurements of the remaining rows or columns
of the Mermin-Peres square. Subplot f demonstrates that the singlet state is a common
(1)
(2)
(1)
(2)
(1)
(2)
eigenstate of the observables σx ⊗ σx , σy ⊗ σy , σz ⊗ σz , as only one of the spheres
has a considerable volume. Taking into account all the results, we find hXKS i = 5.46(4) in
this measurement. Error bars are 1σ.
84
7.3 Testing the Kochen Specker inequality
Figure 7.3.: State-independence of the Kochen-Specker inequality. The Kochen-Specker
inequality was tested for ten different quantum states, including maximally entangled (Ψ1 Ψ3 ), partially entangled (Ψ4 ) and separable (Ψ6 -Ψ9 ) almost pure states as well as an entangled mixed state (ρ5 ) and an almost completely mixed state (ρ10 ). All states are analysed
by quantum state tomography which yields for the experimentally produced states Ψ1 -Ψ4 ,
Ψ6 -Ψ9 an average fidelity of 97(2)%. For all states, one obtains a violation of inequality (7.5)
which demonstrates its state-independent character, the dashed line indicating the average
value of hXKS i. Error bars are 1σ (6,600 state realizations per data point).
85
Correlations
1
0.95
0.9
0.85
0.8
0.75
row 1
row 2
row 3
column 1
column 2
column 3
Set of observables
Figure 7.4.: Permutation within rows and columns of the Mermin-Peres square. As the
three observables of a set are commuting, the temporal order of their measurements should
have no influence on the measurement results. The figure shows the measured absolute values
of the products of observables for any of the six possible permutations. The scatter in the
experimental data is caused by experimental imperfections that affect different permutations
differently. For the measurements shown here, in total 39,600 copies of the singlet state were
used.
inequality hXKS i fall within the range of 5.22 to 5.49. Because of experimental imperfections,
the experimental violation of the Kochen-Specker inequality falls short of the quantum-mechanical
prediction. The dominating error sources are imperfect unitary operations, in particular the entangling gates applied up to six times in a single experimental run. The errors are discussed further
in section 7.5.1
7.4. Including imperfect measurements to close loopholes
All experimental tests of hidden-variable theories are subject to various possible loopholes. In
this experiment, the detection loophole does not play a role, as the state of the ions is detected
with near-perfect efficiency. From the point of view of a hidden variable theory, still objections
can be made: In the experiment, the observables are not perfectly compatible and since the
observables are measured sequentially, it may be that the hidden variables are disturbed (DHV)
during the sequence of measurements, weakening the demand to assign to any observable a fixed
value independently of the context.
Nevertheless, it is possible to derive inequalities for classical non-contextual models, wherein
the hidden variables are disturbed during the measurement process [160]. The starting point to
include imperfections is a Clauser-Horne-Shimony-Holt (CHSH) type inequality
hXCHSH i = hABi + hCDi + hACi − hBDi
(7.8)
one can derive from equation (7.5) by replacing array (7.3) by
I (1) ⊗ I (2)
A = I (1) ⊗ σy
(2)
B=
σy ⊗ σy
I (1) ⊗ I (2)
C = σx ⊗ I (2)
D=
σx ⊗ σx
I
(1)
⊗I
(2)
(1)
I
(1)
⊗I
86
(2)
I
(1)
(2)
(1)
(2)
(1)
⊗I
(2)
.
(7.9)
7.4 Including imperfect measurements to close loopholes
All nine observables fulfill the condition of compatibility but the inequality, similar to Bell tests,
is now violated only by specific states. As most of the observables are now equal to the identity
matrix the relevant measurements are A, B, C, D. A noncontextual HV model has to assign a fixed
value to each measurement, and any such model predicts
| hXCHSH i | ≤ 2.
(7.10)
Ψ ∝ | ↑↑i + iγ| ↓↑i + γ| ↑↓i + i| ↓↓i
(7.11)
If we now consider the QM state
where γ =
√
2 − 1, we will find for equation 7.8 a value of
√
hXCHSH i = 2 2
(7.12)
violating the HV inequality. The choice of observables is not unique and one can transform all
observables by a global unitary transformation and obtain another set. In fact transformations are
possible that lead to a maximal violation for unentangled states.
To deal with the case of imperfect measurements consider a HV model with a probability distri+
+
bution p(λ) and let p[(A+
1 |A1 ) and (B1 |B1 )] denote the probability of finding A if A is measured
first and B + if B is measured first, where these conditions are put onto the same instance of the
HV. This probability is well defined in all HV models of the considered type but it is impossible
to experimentally measure it directly. The aim is now to connect it to probabilities arising in
sequential measurements, as this allows one to find contradictions between HV models and QM.
First, note that
+
+
+
+
−
p[(A+
1 |A1 ) and (B1 |B1 )] ≤ p[A1 , B2 |A1 B2 ] + p[(B1 |B1 ) and (B2 |A1 B2 )].
(7.13)
+
This inequality is valid because if λ is such that it contributes to p[(A+
1 |A1 ) and (B1 |B1 )], then ei+
ther the value of B stays the same when measuring A1 B2 (hence λ contributes to p[A+
1 , B2 |A1 B2 ])
+
−
or the value of B is flipped and λ contributes to p[(B1 |B1 ) and (B2 |A1 B2 )]. The first term
+
p[A+
1 , B2 |A1 B2 ] is directly measurable as a sequence, but the second term is not directly experimentally accessible.
Let us rewrite
−
−
+
hABi = 1 − 2p[(A+
1 |A1 ) and (B1 |B1 )] − 2p[(A1 |A1 ) and (B1 |B1 )],
(7.14)
±
as the mean value obtained from the probabilities p[(A±
1 |A1 ) and (B1 |B1 )]. Then, using equation (7.13), it follows that
hA1 B2 i − 2pflip [AB] ≤ hABi ≤ hA1 B2 i + 2pflip [AB],
(7.15)
where we used pflip [AB] = p[(B1+ |B1 ) and (B2− |A1 B2 )] + p[(B1− |B1 ) and (B2+ |A1 B2 )] which can be
interpreted as a probability that A flips a predetermined value of B.
87
Furthermore, using (7.8), we obtain within a HV model
| hXCHSH i | ≤ 2(1 + pflip [AB] + pflip [CD] + pflip [AC] + pflip [BD]).
(7.16)
The terms pflip [AB], etc. in inequality (7.16) are not experimentally accessible. Now we will discuss
how they can be experimentally estimated when some assumptions on the HV model are made.
In order to obtain an experimentally testable version of inequality (7.16), we will assume that
p [(B1+ |B1 ) and (B2− |A1 B2 )] ≤ p [(B1+ |B1 ) and (B1+ , B3− |B1 A2 B3 )] = p [B1+ , B3− |B1 A2 B3 ]. (7.17)
We must stress at this point that we do not assume that the set of HV values giving [(B1+ |B1 )
and (B2− |A1 B2 )] is contained in the set giving (B1+ , B3− |B1 A2 B3 ), the assumption only relates
the sizes of these two sets. The motive is the following experimental procedure: Let us assume that
one has a physical state, for which one surely finds B1+ , if B1 is measured first, but one finds B2−
if the sequence A1 B2 is measured. Physically, one would explain this behavior with a disturbance
of the system due to the experimental procedures made when measuring A1 . The left hand side
of (7.17) can be viewed as the amount of this disturbance. The right hand side quantifies the
disturbance of B when the sequence B1 A2 B3 is measured. In real experiments, it can be expected
that this disturbance is larger than when measuring A1 B2 , because of the additional experimental
procedures involved. Note that in real experiments, a measurement of B will also disturb the value
of B itself, as can be seen from the fact that sometimes the values of B1 and B2 will not coincide,
if the sequence B1 B2 is measured (see chapter 7.5.1).
Assumption (7.17) gives a measurable upper bound on pflip [AB]. One directly has
|hXDHV i| = |hXCHSH i| − 2(perr [B1 A2 B3 ] + perr [D1 C2 D3 ]+
+ perr [C1 A2 C3 ] + perr [D1 B2 D3 ]) ≤ 2,
(7.18)
where we used perr [B1 A2 B3 ] = p[B1+ , B3− |B1 A2 B3 ] + p[B1− , B3+ |B1 A2 B3 ], denoting the total disturbance probability of B when measuring B1 A2 B3 .
The point with this inequality is that if the observable pairs (A, B), (C, D), (A, D), and (B, D)
fulfill approximately the condition (i) of definition 1 in the definition of compatibility, the terms
perr will become small, and a violation of inequality (7.18) can be observed. To test this inequality
we prepared the state Ψ and performed the measurements listed in (7.9). We find for the value
hXDHV i = 2.23(5) > 2. This proves that even disturbances of the hidden variables for not perfectly
compatible measurements cannot explain the given experimental data. In principle, our analysis
of measurement disturbances and dynamical hidden variable models can be extended to the full
Mermin-Peres square. The error terms considering now four measurements in a row look like
pflip [(AB)C] ≤ perr [(AB)C] = p[C1+ C4− |C1 A2 B3 C4 + p[C1− C4+ |C1 A2 B3 C4 ].
(7.19)
Then, if one writes down the generalized form of inequality (7.5) there are more correction terms
than in inequality (7.18), moreover they occur now with weight 4. On average, these perr terms
have to be smaller than 2/48 ≈ 0.0417 in order to allow a violation. As can be seen in the next
88
7.5 Experimental results on imperfect measurements
Table 7.2.: Measurement correlations hAi Aj |A1 . . . A5 i between repeated measurements of
A = σz ⊗ I for a maximally mixed state. Observing a correlation of hAi Aj |A1 . . . A5 i = αij
means that the probability for the measurement results of Ai and Aj to coincide equals
(αij + 1)/2.
αij
1
2
3
4
2
0.97(1)
3
0.97(1)
0.97(1)
4
0.96(1)
0.97(1)
0.98(1)
5
0.95(1)
0.96(1)
0.98(1)
0.98(1)
Table 7.3.: Measurement correlations hAi Aj |A1 . . . A5 i between repeated measurements of
A = σx ⊗ σx for a maximally mixed state. Observing a correlation of hAi Aj |A1 . . . A5 i = αij
means that the probability for the measurement results of Ai and Aj to coincide equals
(αij + 1)/2.
αij
1
2
3
4
2
0.94(1)
3
0.88(1)
0.93(1)
4
0.82(2)
0.87(2)
0.90(1)
5
0.80(2)
0.84(2)
0.87(2)
0.93(1)
section the experimental techniques have to be improved to get in this regime and find a violation.
7.5. Experimental results on imperfect measurements
The fact that the results falls short of the quantum mechanical prediction of hXKS i = 6 is, as
pointed out, due to imperfections in the measurement procedure. These imperfections could be
incorrect unitary transformations but also errors occurring during the fluorescence measurement.
An instructive test consists in repeatedly measuring the same observable on a single quantum
system and analyzing the measurement correlations. Table 7.2 shows the results of five consecutive
measurements of A = σz ⊗ I on a maximally mixed state based on 1100 experimental repetitions.
As expected, the correlations αi,j = hAi Aj |A1 . . . A5 i for j=i+1 are independent of the measurement number i within the error bars. However, the correlations αi,j for j=i+k become
smaller and smaller the bigger k gets. Table 7.3 shows another set of measurements correlations hAi Aj |A1 . . . A5 i where A = σx ⊗ σx . Here, the correlations are smaller, since entangling
interactions are needed for mapping A onto σz ⊗ I which is experimentally the most demanding
step.
It is also interesting to compare the correlations hA1 A3 |A1 A2 A3 i with the correlations hA1 A3 |A1 B2 A3 i
for an observable B that is compatible with A. For A = σx ⊗ σx and B = σz ⊗ σz , we find
hA1 A3 |A1 A2 A3 i = 0.88(1) and hA1 A3 |A1 B2 A3 i = 0.83(2) when measuring a maximally mixed
state, i. e. it seems that the intermediate measurement of B perturbs the correlations slightly
more than an intermediate measurement of A. Similar results are found for a singlet state,
where hA1 A3 |A1 A2 A3 i = 0.92(1), hB1 B3 |B1 B2 B3 i = 0.91(1), but hA1 A3 |A1 B2 A3 i = 0.90(1),
and hB1 B3 |B1 A2 B3 i = 0.89(1). Because of hB1 B3 |B1 A2 B3 i = 1 − 2perr (B1 A2 B3 ), correlations of
the type hB1 B3 |B1 A2 B3 i are required for checking the inequality (7.18) that takes into account
disturbed HVs.
89
7.5.1. Experimental limitations
There are a number of error sources contributing to imperfect state correlations, the most important
being:
(i) Wrong state assignment based on fluorescence data. As described in chapter 4.5 we use a
250 µs detection time to determine the state of the ion. The photon count distributions slightly
overlap. Therefore, if the threshold for discriminating between the dark and the bright state is set
properly the probability for wrongly assigning the quantum state is 0.3%. Making the detection
period longer would reduce this error but increase errors related to decoherence of the other ion’s
quantum state that is not measured.
(ii) Imperfect optical pumping. During fluorescence detection, the ion leaves the computational
subspace {|0i, |1i} if it was in state |1i and can also populate the state |S1/2 , mS = −1/2i. To
prevent this leakage, the ion is briefly pumped on the S1/2 ↔ P1/2 transition with σ+ -circularly
polarized light to pump the population back to |1i. Due to imperfectly set polarization and
misalignment of the pumping beam with respect to the quantization axis, this pumping step fails
with a probability of about 0.5%.
(iii) Interactions with the environment. Due to the non-zero differential Zeeman shift of the
ion’s states used for storing quantum information, superposition states dephase in the presence of
slowly fluctuating magnetic fields. In particular, while measuring one ion by fluorescence detection,
quantum information stored in the other ion dephases. We partially compensate for this effect by
spin-echo-like techniques [23] that are based on a transient storage of superposition states in a pair
of states having an opposite differential Zeeman shift as compared to the states |0i and |1i1 . A
second interaction to be taken into account is spontaneous decay of the metastable state |0i which
however only contributes an error of smaller than 0.1%.
(iv) Imperfect unitary operations. The mapping operations are not error-free. This concerns
in particular the entangling gate operations needed for mapping the eigenstate subspace of a spin
correlation σi ⊗σj onto the corresponding subspaces of σz ⊗I. For this purpose, a Mølmer-Sørensen
gate operation U M S (π/2, φ) is used. This gate operation has the crucial property of requiring the
ions only to be cooled into the Lamb-Dicke regime. In the experiments, the center-of-mass mode
used for mediating the gate interaction is in a thermal state with an average of 18 vibrational
quanta. In this regime, the gate operation is capable of mapping |11i onto a state |00i + eiφ |11i
with a fidelity of about 98% (see chapter 6.1). Taking this fidelity as being indicative of the gate
fidelity, one might expect errors of about 4% in each measurement of spin correlations σi ⊗ σj as
the gate is carried out twice, once before and once after the fluorescence measurement.
These error sources prevented us from testing a generalization of inequality (7.5). A measurement
of the correlations hB1 B3 |B1 A2 B3 i and hC1 C4 |C1 A2 B3 C4 i resulted in error terms perr that were
about 0.06 for sequences involving three measurements and about 0.1 for sequences with four
measurements, i. e. more than twice as big as required for observing a violation of (7.5).
1 The
state of the ion not to be measured is stored in a superposition of D5/2 , m = 3/2 and D5/2 , m = 5/2 during
the detection. After remapping the state to D5/2 , m = 3/2 and S1/2 , m = 1/2 a waiting time is implemented
during which the states partially rephase.
90
8. Quantum simulation of the Dirac
equation
Quantum simulation aims at simulating a quantum system using a controllable laboratory system
with the same underlying mathematical model. In this way, it is possible to simulate quantum
systems that can neither be efficiently simulated on a classical computer [2] nor easily accessed
experimentally, while allowing parameter tunability over a wide range.
Here, a proof-of-principle quantum simulation of the one-dimensional Dirac equation using a
single trapped ion [117] is demonstrated. The ion is set to behave as a free relativistic quantum
particle. The particle position was measured as a function of time and Zitterbewegung was studied
for different initial superpositions of positive and negative energy spinor states, as well as the
cross-over from relativistic to nonrelativistic dynamics. The high level of control of trapped-ion
experimental parameters makes it possible to simulate elegantly textbook examples of relativistic
quantum physics.
8.1. Introduction
The difficulties in observing real quantum relativistic effects have sparked significant interest in
performing quantum simulations of their dynamics. Examples include black holes in Bose-Einstein
condensates [163], and Zitterbewegung for massive fermions in solid state physics [164] none of
which have been experimentally realized so far. Graphene is studied widely in connection to the
Dirac equation [165–167].
The Dirac equation is a cornerstone in the history of physics [168], merging successfully quantum mechanics with special relativity, providing a natural description of the electron spin and
predicting the existence of anti-matter [169]. Furthermore, it is able to reproduce accurately the
spectrum of the hydrogen atom and its realm, relativistic quantum mechanics, is considered as
the natural transition to quantum field theory. However, the Dirac equation also predicts some
peculiar effects such as Klein’s paradox [170] and Zitterbewegung, an unexpected quivering motion
of a free relativistic quantum particle first examined by Schrödinger [171]. These and other predictions would be difficult to observe in real particles, while constituting key fundamental examples
to understand relativistic quantum effects. Recent years have seen an increased interest in simulations of relativistic quantum effects in different physical setups [117, 163, 164, 172–175], where
parameter tunability allows accessibility to different physical regimes.
Trapped ions are particularly interesting for the purpose of quantum simulation [176–178], as
they allow for exceptional control of experimental parameters, and initialization and read-out
can be achieved with high fidelity. Recently, for example, a proof-of-principle simulation of a
91
Quantum simulation of the Dirac equation
quantum magnet has been performed [63] using trapped ions. The full three-dimensional Dirac
equation Hamiltonian can be simulated with lasers coupling to the three vibrational eigenmodes
and the internal states of a single trapped ion [117]. The setup can be significantly simplified when
simulating the Dirac equation in 1+1 dimensions.
8.2. The Dirac equation
This section closely follows reference [179] to explain the effects observed in the experiment. The
Dirac equation for a spin-1/2 particle with rest mass m is given by
i~
∂ψ
= (c α · p̂ + βmc2 )ψ.
∂t
(8.1)
Here c is the speed of light, p̂ is the momentum operator, α and β are the Dirac matrices, which are
usually given in terms of the Pauli matrices σi [168] and the Identity I2 (as defined in chapter 2.1.2)
Ã
αi =
0
σi
σi
0
!
Ã
, β=
I2
0
0
−I2
!
.
(8.2)
Each of the entries in these matrices is again a 2 x 2 matrix with i = x, y, z for the three components
of the momentum operator. The wave functions ψ are 4-component spinors related on one hand
to the spin of the particle and on the other hand to positive and negative energy solutions. The
original idea by Dirac was that the negative energy states correspond to anti-particles while positive
energy states correspond to the particle. The problem with this interpretation is that by radiative
interaction positive energy states could be transformed into negative energy states and atoms
would not be stable. Again Dirac proposed a possible solution [180]. All negative energy states
are filled such that no transition is possible due to the Pauli exclusion principle. If one of these
negative energy electrons is excited it will leave a hole in this sea of particles with a positive charge
representing the anti particle. This interpretation has to be handled with care as a number of
problems appear. The ground state has an infinitely high negative energy. There is an asymmetry
between particles and anti-particles and we have a multi particle system. A complete solution of
these problems is only achieved by using a full quantum electro-dynamics theory.
To describe the realized experiment we restrict ourselves to the 1+1 dimensional case of the
Dirac equation
∂ψ
i~
= HD ψ = (c p̂ σx + mc2 σz )ψ.
(8.3)
∂t
This restriction does not affect the appearance of the most stunning effects of the Dirac equation,
such as Zitterbewegung and the Klein paradox. The two components of the spinor in 1+1 dimension
do not describe the spin as in one dimension all magnetic fields are pure gauge fields [179] and spin
does not exist. They rather represent the appearance of positive and negative energy states
Ã
Ψ(x, t) =
Ψ1 (x, t)
Ψ2 (x, t)
92
!
.
(8.4)
8.2 The Dirac equation
For a further investigation the Dirac Hamiltonian is rewritten in matrix form
Ã
HD =
This matrix has two eigenvalues E(p) = ±
solutions to the Dirac equation u± (p; x, t)
mc2
cp̂
cp̂
−mc2
p
!
.
(8.5)
p2 c2 + m2 c4 so there are two types of plane wave
1
u± (p; x, t) = √ u± (p)eipx/~∓iE(p)/~t
2π
(8.6)
with u± the eigenvectors to the respective eigenvalues E(p). As a solution for the Dirac equation
we hence have plane waves with positive and negative energy
HD u± (p; x, t) = ±E(p) u± (p; x, t).
(8.7)
To account for a real physical situation, that is to find the particle between x = +∞ and x = −∞
with probability 1, we have to demand that the integral
Z
+∞
(|Ψ1 (x, t)|2 + |Ψ2 (x, t)|2 )dx = 1
(8.8)
−∞
is equal to one. The same argument holds for the Fourier transformed spinors Ψ̂1,2 (p, t) =
F(Ψ̂1,2 (x, t)) reflecting the momentum distribution. The unitary time evolution of the Dirac equation guarantees that the normalization of the Dirac spinor is time-independent. One has to note
though, that this is not necessarily true for the individual components Ψ1,2 . What holds true is,
that if there is a particle at the beginning their will be a particle in the end. The Dirac equation
cannot describe particle creation or annihilation.
Square-integrable wave packets can be obtained by superposition of plane waves
Z
+∞
Ψ(x, t) =
(Ψ̂+ (p) u+ (p; x, t) + Ψ̂− (p) u− (p; x, t))dp.
(8.9)
−∞
The coefficient functions Ψ̂± (p) can be obtained by a projection of Ψ(p, 0) onto the positive and
negative energy subspace
Ψ̂± (p) = P ± (p)Ψ(p, 0).
(8.10)
Ψ(p, 0) is obtained by Fourier transforming the initial wave function Ψ(x, 0). In momentum space,
the projection operators are given by
1
P (p) =
2
±
Ã
cp̂σx + mc2 σz
I2 ± p
c2 p̂2 + m2 c4
!
.
(8.11)
Note that in general the projection operators do not project onto the spinor basis states except for
the case where p = 0. Only in this case the projector (8.11) becomes diagonal in the spinor basis.
In the Dirac equation in 1+1 dimensions, as given in equation (8.3), there is only one motional
degree of freedom and the spinor is encoded in two internal levels, related to positive and negative
93
energy states [117]. The velocity of the free Dirac particle is given by
dx̂/dt = [x̂, HD ]/i~ = cσx
(8.12)
in the Heisenberg picture. For a massless particle [σx , HD ] = 0 and hence σx is a constant of
motion. For a massive particle [σx , HD ] 6= 0, and the evolution of the particle is described by [171]
−1
ˆ 2iHD t/~ − 1),
x̂(t) = x̂(0) + p̂c2 HD
t + i ξ(e
(8.13)
−1
−1
with ξˆ = 12 ~c(σx − p̂cHD
) HD
. The first two terms represent an evolution linear in time, as
expected for a free particle, while the last oscillating term may induce Zitterbewegung.
Zitterbewegung is understood as an interference effect between the positive and negative energy
parts of the spinor and does not appear for spinors that consist entirely out of positive (or negative)
energy. Furthermore, it is only present when these parts have significant overlap in position and
momentum space and is therefore not a sustained effect under most circumstances [168]. For a
free electron, the Dirac equation predicts the Zitterbewegung to have an amplitude of the order
of the Compton wavelength RZB ∼ 10−12 m and a frequency of ωZB ∼ 1021 Hz and has so far
been experimentally inaccessible. The real existence of Zitterbewegung, in relativistic quantum
mechanics and in quantum field theory, has been a recurrent subject of discussion in the last
years [181, 182].
Note that a particle consisting of positive and negative energy states can have zero average
momentum but a non-zero average velocity. The average velocity of a wave packet is given by the
operator known from classical relativistic mechanics (see equation (8.13))
−1
v̂CL (p) = c2 p̂ HD
.
(8.14)
This relation depends on the sign of the energy. For a wave packet with negative energy, a negative
momentum corresponds to a positive velocity. Thus for an equal superposition of positive and
negative energy states it is possible that the respective average momenta cancel while the average
velocities are the same. Exactly this situation is simulated in figure 8.1 of the following chapter.
8.3. Experimental realization
For the simulation, a single 40 Ca+ ion is trapped in a linear Paul trap [76] with trapping frequencies
ωax = 2π×1.36 MHz axially and ωrad = 2π×3 MHz in the radial directions. Doppler cooling,
optical pumping, and resolved sideband cooling on the S1/2 ↔ D5/2 transition in a magnetic field
of 4 G prepare the ion in the axial motional ground state and in the internal state |S1/2 , m = 1/2i.
¡¢
¡¢
A narrow linewidth laser at 729 nm couples the states 01 := |S1/2 , m = 1/2i and 10 := |D5/2 , m =
3/2i which we identify as our spinor states. A bichromatic light field resonant with the upper and
¡¢
¡¢
lower axial motional sidebands of the 10 ↔ 01 transition with appropriately set phases [183]
realizes the first part, proportional to σx p̂, of the Hamiltonian (see chapter 3.6)
HD = 2η∆Ω̃σx p̂ + ~Ωσz .
94
(8.15)
8.3 Experimental realization
Figure 8.1.: Expectation values hx̂(t)i for particles with different masses. The linear
curve (¥) represents a massless particle (Ω = 0) moving with the speed of light given by
c = 2η Ω̃∆ = 0.052 ∆/µs for all curves. The other curves are for particles with increasing
mass moving down from the linear curve. Their Compton wavelengths are given by λC :=
2η Ω̃∆/Ω = 5.4∆ (H), 2.5∆ (¨), 1.2∆ (•) and 0.6∆ (N), respectively. The solid curves
represent numerical simulations. The figure shows Zitterbewegung for the crossover from
the relativistic 2η Ω̃ À Ω to the nonrelativistic limit 2η Ω̃ ¿ Ω. The inset shows fitted
Zitterbewegung amplitude RZB (¥) and frequency ωZB (•) versus the parameter Ω/η Ω̃
(which is proportional to the mass). Error bars are 1σ.
The component proportional to σz (equivalent to a light shift) is achieved by detuning the red
p
and blue light field by Ω from resonance1 . ∆ = ~/2m̃ωax is the size of the ground-state wave
function with m̃ the ion’s mass, not to be confused with the mass m of the simulated particle,
†
~
η = 0.06 is the Lamb-Dicke parameter, p̂ = a 2i−a ∆
is the momentum operator and a† and a are
the usual raising and lowering operators for the motional state along the axial direction. The first
term in equation 8.15 describes a state-dependent motional excitation with coupling strength η Ω̃,
corresponding to a displacement of the ion’s wave packet in the harmonic trap. The parameter Ω̃
is controlled by setting the intensity of the bichromatic light field. The second term is equivalent to
an optical Stark shift and arises when the bichromatic light field is detuned from resonance by 2Ω.
Equation (8.15) reduces to the 1+1 dimensional Dirac Hamiltonian if we make the identifications
c := 2η Ω̃∆ and mc2 := ~Ω. The momentum and position of the Dirac particle are then mapped
onto the corresponding quadratures of the trapped ion harmonic oscillator.
In order to study relativistic effects such as Zitterbewegung it is necessary to measure the expectation value of the position operator hx̂(t)i of the harmonic oscillator. It has been noted theoretically
that such expectation values could be measured using very short probe times, without reconstructing the full quantum state [115–117]. As pointed out in chapter 3.6.1, for measuring hx̂i of a
motional state ρm , we have to (i) prepare the ion’s internal state in an eigenstate of σy , (ii) apply
a unitary transformation U (τ ) that maps information about ρm onto the internal states, and (iii)
record the changing excitation by a fluorescence measurement [76] as a function of the probe time
1 Both
light fields are detuned in the same direction unlike a Mølmer-Sørensen interaction.
95
Figure 8.2.: Evolution of the expectation value hx̂(t)i for particles with mass λc = 1.2∆
and an electronic state which is an eigenstate of σx (•) (data taken from figure 8.1) or
an eigenstate of σy (¥). Zitterbewegung appears again due to interference of positive and
negative energy parts of the state. Even though the simulated particles have the same mass
their behavior is completely different. For the σy eigenstate the initial velocity is zero (see
equation 8.12) whereas the particle in an eigenstate of σx has an initial velocity equal to the
speed of light. The solid curve represents a numerical simulation. Error bars are 1σ.
τ (see chapter 3.6.1). In this protocol, the unitary operator U (τ ) = exp(−iηΩp σx x̂τ /∆), with
x̂ = (a† + a)∆ and probe Rabi frequency Ωp , effectively transforms the observable σz into sin kx̂,
with k = 2ηΩp τ /∆, so that hx̂i can be determined by monitoring the rate of change of hsin kx̂i
for short probe times (see chapter 3.6.1). Since the Dirac Hamiltonian generally entangles the motional and internal state of the ion, we first incoherently recombine the internal state population in
¡0¢
2
1 before proceeding to (i). Then, we apply the Hamiltonian generating U with the probe Rabi
frequency set to Ωp = 2π× 13 kHz and for interaction times τ of up to 14 µs, with 1-2 µs steps.
The change of excitation was obtained by a linear fit each based on 104 to 3 · 104 measurements.
We simulate the Dirac equation by applying HD for varying amounts of time and for different
particle masses. In the experiment we set Ω̃ = 2π × 68 kHz, corresponding to a simulated speed of
light c = 0.052 ∆/µs. The measured expectation values hx̂(t)i are shown in figure 8.1 for a particle
√
x2 ¡ ¢
1
initially prepared in the spinor state ψ(x; t = 0) = ( 2π2∆)− 2 e− 4∆2 11 by sideband cooling and
application of a π/2-pulse. Zitterbewegung appears for particles with non-zero mass, obtained by
varying Ω in the range of 0 < Ω ≤ 2π× 13 kHz by changing the detuning of the bichromatic lasers.
We investigate the particle dynamics in the cross-over from relativistic to nonrelativistic dynamics. The data in figure (8.1) match well with numerical simulations based on the Hamiltonian (8.15)
` ´
is done by first shelving the population initially in 01 to |D5/2 , m = 5/2i using a Rapid Adiabatic Passage
` ´
` ´
transfer (RAP). A second RAP transfers the population in 10 to 01 . A 100 µs laser pulse at 854 nm transfers
` ´
the population in |D5/2 , m = 5/2i to |P3/2 , m = 3/2i from which it spontaneously decays to 01 . The transfer
efficiency is > 99 %, limited by the small branching ratio to the D3/2 state. In the transfer steps a probability
exists that the motional state of the ion is changed. This probability is however very small due to the small
Lamb-Dicke parameter
` ´ but
` ´could be eliminated completely by a separate measurement of the motional states of
the spinor states 01 and 10 , at the expense of a longer data acquisition time.
2 This
96
Figure 8.3.: (a) Zitterbewegung for a state with non-zero average momentum. Initially,
Zitterbewegung appears due to interference of positive and negative energy parts of the
state. As these parts separate, the oscillatory motion fades away. The solid curve represents
a numerical simulation. (b) Numerically calculated probability distributions |ψ(x)|2 at the
times t = 0, 75 and 150 µs ¡(as
¢ indicated by the arrows in (a)). The probability distribution
corresponding to the state 01 is inverted for clarity. The vertical solid line represents hx̂i as
plotted in (a). The two dashed lines are the expectation values for the positive and negative
energy parts of the spinor. (c) Time evolution for a negative energy eigenstate with λC =
1.2∆. Laser pulses create a negative energy spinor with average momentum hp̂i = 2.2~/∆.
The corresponding initial momentum distribution |ψ̃(p)|2 is shown in the inset. The solid
lines represent a numerical calculation. The curve in (c) shows no Zitterbewegung. (d)
Simulated probability distributions |ψ(x)|2 for three different evolution times are shown (as
indicated by the arrows in (c)). Error bars are 1σ.
shown as solid lines. The error bars are obtained from a linear fit assuming quantum projection
noise which dominates over noise caused by fluctuations of control parameters. In addition, the
data were fitted with a heuristic model function of the form hx̂(t)i = at + RZB sin ωZB t to extract
the effective amplitude RZB and frequency ωZB of the Zitterbewegung shown in the inset. As the
particle’s initial momentum is not dispersion-free, the amplitude and frequency are only approximate concepts. From these data it can be seen that the frequency grows linearly with increasing
mass ωZB ≈ 2Ω, whereas the amplitude decreases as the mass is increased. Since the mass of
the particle is increased but the momentum and the simulated speed of light remain constant, the
data in figure (8.1) show the crossover from the far relativistic to nonrelativistic limits. Hence,
the data confirm that Zitterbewegung vanishes in both limits, as theoretically expected. In the
far relativistic case this is because ωZB goes to zero, and in the nonrelativistic case because RZB
vanishes.
The tools used for simulating the Dirac equation can also be applied to precisely set the initial
state of the simulated particle. A first simple change is to prepare an electronic eigenstate of σy .
√
x2 ¡ ¢
1
The wave function of the particle is given by ψ(x; t = 0) = ( 2π2∆)− 2 e− 4∆2 1i . The evolution of
97
Figure 8.4.: (a) Blue sideband oscillations after a 100 µs application of Hamiltonian (8.15)
with Ω = 0 and Ω̃ = 2π × 68 kHz demonstrating the coherence of the displacement. The
oscillations for the coherent state with hni = 6.3 die out quickly and revive after 500 µs.
The solid line was obtained by simulating the phonon distribution and coherently adding the
individual state evolutions. (b) Blue sideband oscillations for the same parameters except
λc = 1.2∆. The solid line was again obtained by simulating the phonon distribution and
coherently adding the individual state evolutions.
the expectation values hxi shown in figure 8.2 for particles with equal mass λc = 1.2∆ but different
electronic state is quite different. The particle initially in an eigenstate of σy has zero velocity at
t=0 (see equation (8.12)) whereas the particle in an eigenstate of σx has an initial velocity equal
to the speed of light.
The particle in figure 8.3a was given an average initial momentum hp̂(t = 0)i = ~/∆ by a
displacement operation with the Hamiltonian H = ~η Ω̃σx x̂/∆. The initial state of this particle is
built up out of a positive energy component with positive velocity and a negative energy component
with negative velocity[179]. The positive energy component moves to the right and both spinor
states can be seen to contribute to this part (see appendix B.1), the negative component moves
to the left. As long as these parts overlap, Zitterbewegung is observed which dies out as the parts
separate. For an additional illustration how the vanishing of the Zitterbewegung comes about the
evolution of the probability distributions are shown in figure 8.3 b. The solid lines are obtained by
¡¢
numerical simulations. The probability distribution corresponding to the state 01 is inverted for
clarity.
It is also possible to initialize the spinor in a pure negative or positive energy state (see appendix B.1). In figure 8.3 c, the time evolution hx̂(t)i of a negative energy spinor with average
momentum hp̂i = 2.2~/∆ is shown. The corresponding simulated probability distributions are
displayed in figure 8.3 d and it can be seen that there is indeed no Zitterbewegung or splitting of
the wave function which arises only if there are positive and negative energy contributions to the
wave function.
An additional test to compare experimentally obtained states with theory can be performed by
an excitation on the blue sideband after recombining the internal state population. An excitation
on the blue sideband will lead to oscillations that die out quickly but revive after some time
due to an interference of the coherently populated states |↓, ni. This interference occurs because
all phonon states are excited from the S1/2 ground state to the D5/2 excited state at the same
98
time however their Rabi frequency is different. The populations will thus oscillate at different
frequencies between ground and excited state and the sum of these oscillations is quickly decaying.
Figure 8.4 (a) shows oscillations on the blue sideband after a 100 µs application of Hamiltonian 8.15
√
x2 ¡ ¢
1
acting on the initial spinor state ψ(x; t = 0) = ( 2π2∆)− 2 e− 4∆2 11 . The parameters in the
Hamiltonian where set to Ω = 0 and Ω̃ = 2π × 68 kHz. The displacement created a coherent state
with hni = 6.3 corresponding to α = 2.5. Similar curves have already been observed in David
Winelands group [184]. Figure 8.4 (b) shows blue sideband oscillations for the same initial state
and parameters, except Ω = 2π ×7 kHz. The solid lines in both figures were obtained by simulating
the phonon distribution and summing over it,
ρDD (t) =
∞
X
pn sin2 (Ωn,n+1 t),
(8.16)
n=0
with Ωn,n+1 the Rabi frequency for coupling the motional states n and n+1 and pn the occupation
probability of the motional state |ni.
99
100
9. Summary and Outlook
The main results of this thesis can be divided into three topics: QIP with 40 Ca+ and 43 Ca+
ions realizing high fidelity entangling operations, a test of a hidden variable theory where the test
procedure relies on QIP techniques and finally the quantum simulation of the Dirac equation. All
experiments encode the quantum information in Zeeman levels of the S1/2 ground state and the
meta-stable D5/2 state of calcium ions.
An entangling gate operation is necessary to meet the DiVincenzo criteria to realize a quantum
computer. For this purpose a Mølmer-Sørensen entangling operation was implemented creating
Bell states with 40 Ca+ ions with a fidelity of 99.3%. Assuming that the Bell state fidelity is a good
measure for the gate-operation, the errors are in principle low enough to be in the fault tolerant
regime, although the overhead in the number of qubits is enormous. A detailed experimental
investigation of the gate mechanism identified possible error sources. The high quality of the gate
operation allowed for concatenations of up to 21 individual gate-operations. Furthermore it was
shown for the first time, that this kind of gate operation does not depend on ground state cooling.
Bell states with a fidelity of 98.4(2)% were realized with ions cooled to a thermal state with a mean
vibrational quantum number hni = 18(2).
Implementing the gate operation with three ions was the next obvious step. GHZ states were
created with a fidelity of 97.9(2)%. This is so far the highest achieved fidelity for a three ion
GHZ state. By using an off-resonant tightly focused beam a selective entanglement of two out
of three ions was achieved. The off-resonant beam was used to switch on and off the MølmerSørensen interaction on each qubit individually. This scheme is an important building block for
realizing NMR like techniques in ion traps. The ideas developed in Volckmar Nebendahls [150]
thesis promise simple algorithms realizing e.g. error correction with 9 pulses or a Toffoli gate with
11 pulses.
The knowledge gained with 40 Ca+ was transferred to the more complicated level structure of
43
Ca+ . A clever choice of polarization and geometrical dependencies helped to suppress the coupling strength of unwanted transitions. This paved the way to create entangled states using 43 Ca+
ions with a fidelity of 96.9(3)% despite the presence of spectator states. To protect the susceptible
quantum state against magnetic field fluctuations it was mapped onto the hyperfine clock state
of 43 Ca+ . The mapping pulses work with a fidelity of better than 99% such that the fidelity of
the Bell state encoded in the hyperfine qubits was 96.7%. After a storage time of 20 ms the state
was mapped back to the optical qubit and two additional entangling operations were applied. The
fidelity after this sequence was still 87.8%. This result demonstrates the combination of a high
fidelity gate operation on an optical qubit with the long coherence times of a hyperfine qubit in a
single realization.
The improvements that have to be incorporated for future QIP experiments are manifold. One
101
Summary and Outlook
of the main goals is to unify all building blocks: high fidelity gates, long coherence times, high
fidelity state initialization and read out, in one scalable system. The focus of current technology
development will lie on micro-fabricated traps that provide an environment for ions suitable to
do QIP. Along this development progress has to be made on many aspects. All techniques have
to be modified such that they work reliably on a bigger number of ions at the same time. All
experimental parameters have to be controlled with higher precision e.g. laser intensity, trap
frequencies and magnetic field. Achieving this control of parameters is quite challenging. It might
be more fruitful to develop and improve alternative methods like coherent control [185] or an
encoding in decoherence free subspaces [52] that are more robust against parameter fluctuations.
As it turned out, ion trap systems are a good test bed for fundamental questions in quantum
physics or rather hidden variable theories. For this purpose measurements of the form σi ⊗ σi were
realized by mapping the state of two qubits on a single qubit prior to the fluorescence measurement. The non-demolition measurement was completed by a second operation mapping the state
projected on |↑i or |↓i onto an eigenstate of the measurement. This technique allowed us to do
a series of quantum non-demolition measurements on a single quantum state. The investigated
high fidelity entangling gate working on thermal ions was a necessary prerequisite for this task.
This measurement technique was the key ingredient to do, for the first time, a state-independent
experimental test of the Kochen Specker theorem. A violation of the Kochen Specker inequality
was found with hXKS i = 5.46(4) > 4 indicating that hidden variable theories cannot explain the
observed effects. The theory predicts that this violation will be found for all possible quantum
states. This prediction was tested by performing the same measurements on 10 different input
states, finding a violation for all states. A possible loophole, incompatibility of measurements,
was closed by carefully evaluating possible error sources and including them in a refined theory.
We found that quantum mechanics violates a CHSH like inequality including the errors such that
hXDHV i = 2.23(5) > 2. For a violation of the original Kochen Specker inequality incorporating
imperfect measurements the errors of the mapping operations have to be improved by more than
a factor of two.
Due to the huge success of ion traps with QIP one tends to forget that an ion trap is “the“
model system for a quantum harmonic oscillator. In QIP applications it serves as a mere mediator
to couple different ions, not exploiting the full available Hilbert space. In this thesis the harmonic
oscillator was utilized to realize a simulation of the Dirac equation. Momentum and position of
a free Dirac particle were mapped onto the respective observables of the trapped ion harmonic
oscillator. The exact Dirac Hamiltonian was implemented and effects like Zitterbewegung could be
measured. A new method was used to determine the average position of the wave packets. The
amount of control of the system allowed for a preparation of specific states where the form of the
Zitterbewegung is changed, Zitterbewegung dies out or does not occur at all.
As it turned out the observable A(t) = cos(2ηΩp t/∆x̂)σz + sin(2ηΩp t/∆x̂)σy used for measuring
hx̂i can be adopted to reconstruct the probability distribution of the wave function [78, 186]. This
possibility increases the information about the evolution of the system. It nicely visualizes the
splitting of the wave packet and will be a valuable tool for future simulations.
Possible extensions of the Dirac equation simulation are to add more spatial dimensions or to
incorporate potentials to simulate e.g. Klein’s paradox. An interesting example is a Dirac particle
102
on a slope. The Hamiltonian in 1+1 dimensions reads as
Hpot = cp̂σx + mc2 σz + cax̂I = HD + cax̂I.
(9.1)
The additional potential cax̂ could be realized with a second ion in the trap. HD will be implemented with the first ion using the very same light fields as already demonstrated. A displacement
operator on the second ion working on different electronic levels but on the same motional mode
creates the desired potential. All that is needed in addition to the presented work is a second
bichromatic light field tuned to a different transition. Simulations suggest that a particle running
up this slope could tunnel through the potential by switching to the negative energy eigenstate.
This eigenstate sees an inverted slope and is thus accelerated away from the turning point into the
classically forbidden region. This effect corresponds to Klein’s paradox. Another proposal that
could be simulated using similar interactions is the 2+1 dimensional Dirac oscillator [172].
Several other proof of principle quantum simulation experiments have been demonstrated in
the last years [63, 186, 187]. So simulations of simple quantum systems are already feasible in
ion traps. Besides the already realized experiments a huge variety of quantum simulations has
been proposed [178] for trapped ions. Some examples are the Unruh effect [188, 189], spin models [177], spin boson models [190], cosmological particle creation [191] and the Frenkel-Kontorova
model [192]. It seems that in the near future more complicated experimental realizations with a
few tens of ions are within reach. Such problems would already be hard to simulate with classical
computers.
103
Summary and Outlook
104
A. The Kochen Specker measurements
A.1. Detailed results for the Kochen Specker measurements
The measurement correlations shown in figure 7.2 are based on experiments where measurements
on a total of 6,600 realizations of the quantum state Ψ = √12 (| ↑↓i − | ↓↑i) were performed. In
these experiments, the results were observed (v1 , v2 , v3 ), vi ∈ {−1, +1}, with the frequencies listed
in the table.
Observables
Measured
correlations
Measured
correlations
(2)
(2)
(1)
M3
σx
(1)
σx
(2)
(1)
σx ⊗ σz
σz ⊗ σx
(2)
(1)
σx ⊗ σz
(2)
(1)
σy ⊗ σy
(−1, −1, −1)
17
3
20
(+1, −1, −1)
(−1, +1, −1)
(+1, +1, −1)
520
511
8
516
529
18
495
511
8
(−1, −1, +1)
(+1, −1, +1)
9
4
15
7
23
16
(−1, +1, +1)
(+1, +1, +1)
13
18
10
2
11
16
1100
1100
1100
Number of experiments
Observables
(1)
σz
(2)
σz
(2)
(1)
σz ⊗ σz
M1
M2
(1)
(2)
(2)
(1)
M1
M2
M3
σz
(2)
σx
(2)
(1)
σz ⊗ σx
σz
(1)
σx
(2)
(1)
σx ⊗ σz
σz ⊗ σz
(2)
(1)
σx ⊗ σx
(2)
(1)
σy ⊗ σy
(−1, −1, −1)
11
13
1010
(+1, −1, −1)
(−1, +1, −1)
(+1, +1, −1)
(−1, −1, +1)
(+1, −1, +1)
281
250
11
237
16
298
276
18
264
13
8
14
9
26
11
(−1, +1, +1)
(+1, +1, +1)
17
277
14
204
22
0
1100
1100
1100
Number of experiments
105
The Kochen Specker measurements
Figure A.1.: Sequence used in the Kochen Specker measurements to hide one ion and
detect the state of the other ion. The waiting time counteracts stepwise magnetic field
fluctuations.
A.2. Hiding the second ion in the Kochen Specker
measurements
The QND measurements performed in chapter 7 required the hiding of the S-state of the second
ion from the fluorescence laser in the D5/2 , mj = 5/2 state. The hiding was done by using a pulse
sequence similar to the one shown in figure 5.4(b) using an addressed AC stark pulse. The pulse
(2)
(1)
sequence is given by Ux (π) = Ux ( π2 )Uz (π)Ux ( π2 ). The un-hiding after the detection is done in
the same way. The phases of the two carrier pulses φ that un-hide the ion have to be corrected for
the phase difference the state D5/2 , mj = 5/2 acquires relative to the S1/2 , mj = 1/2 state. The
whole pulse sequence for a Kochen Specker measurement including the detection, un-hiding and
unitary transformations can be seen in figure A.1.
One problem we were facing in these measurements was a slow sometimes stepwise variation
of the magnetic field. The slow variations are tracked by our automatic measurement scheme
determining the magnetic field and laser drift (see chapter 5.3). If a step occurs in the magnetic
field by ∆B then the carrier pulses are off-resonant for some time until the change is detected (in
the worst case this can be up to a minute). Due to the detuning the unitary operations following
the un-hiding will have the wrong phase. This problem can be reduced by the waiting time in the
pulse sequence.
The phase difference the qubit acquires when stored in the D-states is given by
φDD0 = µB · gD · ∆B · ∆ t
(A.1)
(see equation 2.41) where ∆t is the time needed for state detection and optical pumping. The
phase the qubit acquires during the waiting time ∆t0 is given by
φSD = µB · (gS ·
1
3
− gD · )∆B · ∆t0 .
2
2
(A.2)
The total phase the qubit acquires is thus given by φtotal = φSD + φDD0 . It turns out that this
phase can be canceled by choosing
∆t0 =
gD
∆t.
3/2gD − 1/2gS
(A.3)
This trick only works if the laser is more stable than the magnetic field as the length of the whole
pulse sequence is increased. The best compromise in our case was achieved by choosing ∆t = ∆t0
which results in a partial compensation of the phase.
106
B. Methods for simulating the Dirac
equation
B.1. Constructing a pure negative energy spinor
The spinor state in figure 8.3 c was engineered backwards by projecting out the negative energy
part of a wavepacket with average momentum hp̂i = 2.2~/∆ and renormalizing the spinor.
The complete sequence for approximating the negative energy state is conveniently described
¡1¢
in the basis of the eigenstates |±iy = √12 ±i
of σy . After ground state cooling we prepare
the state |+iy , then we displace this state to an average momentum state hp̂i = 2.2~/∆ by the
displacement Hamiltonian H = ~η Ω̃σy x̂/∆. Next, a far detuned laser pulse rotates the internal
state to 0.84|+iy + i0.53|−iy . The displacement Hamiltonian H = −~η Ω̃σy x̂/∆ shifts these parts
in opposite directions to create the required asymmetry between the average momenta of the
components. A final π/2-pulse creates the state shown in figure 8.3 c. This state has > 99%
overlap with the desired negative energy state.
B.2. Setting the phase of the Displacement pulse
Two phases φ+ , φ− have to be set (see Hamiltonian 3.37) to get the right electronic and motional
interaction in the displacement Hamiltonian (8.3). The sum phase φ+ can be set by the overall
phase of the light field (see figure 4.7 AO2 ). The right value in the massless case (Ω = 0) can
be found by a simple experiment. Applying a Uy (π/2) pulse to the ion we create an eigenstate
of σx . If we then apply Hamiltonian 8.3 with Ω = 0 the electronic state of the qubit should not
change. This was confirmed by a second Uy (π/2) pulse completing the rotation to the excited
state. By scanning the overall phase of the bichromatic light field we can find the phase value that
corresponds to a σx interaction and implements the desired Hamiltonian.
In the case where Ω 6= 0 (simulation of a particle with mass) finding the right phase was a little
bit more complicated. We again perform the same experiment, Uy (π/2) - apply HD - Uy (π/2) and
scan the length of the bichromatic pulse. If the phase is set correctly, we find a change of excitation
as shown in figure B.1. The parameters of the displacement pulse were set to Ω̃ = 2π × 68 kHz and
Ω = 2π× 13 kHz. Comparing these data with simulations (also shown in figure B.1) we found that
the right phase can be determined by varying the phase φ+ and minimizing the excitation after
a 40 µs bichromatic pulse. The difference phase φ− can be set by changing the relative phases of
the RF-signal generators driving AO3 (see figure 4.7). Setting the absolute value of φ− of the red
and blue light field is not necessary as we can define our phase space such that the displacement
we do is along x. To get a displacement along p we have to add an additional phase difference
107
Methods for simulating the Dirac equation
1.0
Excitation to D
5/2
0.9
0.8
0.7
0.6
0
50
100
150
Displacement length (µs)
Figure B.1.: Excitation evolution for a pulse length scan of the bichromatic pulse in the
experiment Uy (π/2) - apply HD - Uy (π/2) (•). The parameters of the bichromatic pulse
were Ω̃ = 2π × 68 kHz and Ω = 2π× 13 kHz. Here the phase of the bichromatic pulse was
set right realizing 8.3. The right phase can be determined by minimizing the excitation after
a 40 µs pulse length. The red line is a simulation based in 8.3.
of 90◦ to φ− while keeping the sum frequency φ+ the same. This 90◦ phase shift of φ− can be
achieved by introducing a switchable delay line in the RF line supplying the blue light field. At
the same time we have to change the phase of the light field by 90◦ to maintain the same value
for φ+ and thus the same electronic interaction. A more sophisticated way to change the motional
interaction in a continuous way from x to p can be achieved by opening up the axial confinement.
When we change the trap frequency by an amount of ∆ we have to replace the operators a and a†
in equation (3.38) by aei∆t and a† e−i∆t . So phase space will start to rotate with respect to the
unchanged laser frequency that serves as a reference oscillator. If we change the axial trapping
potential by ∆, wait for an appropriate amount of time t and switch back to the initial confinement
we have rotated phase space by ∆t.
A fast switching of the axial trap frequency was achieved with the setup used for shifting the
ions (see chapter 4.1) but supplying both tips from one of the outputs of the circuit shown in
figure 4.2. We can change the trap frequency by about 6 kHz which corresponds to a switching
time from x to p of about 40 µs.
108
C. Physical and optical properties of
Calcium
speed of light
permeability of vacuum
permittivity of vacuum
Planck’s constant
elementary charge
Bohr magneton
atomic mass unit
electron mass
fine structure constant
Boltzmann constant
c
µ0
²0
h = 2π~
e
µB
u
me
α
kB
299 792 458 m/s (exact)
4 π · 107 (exact)
1/(µ0 c2 ) (exact)
6.626 068 96(33) ·10−34 J s
1.602 176 487(40) ·10−19 C
927.400 915(23) ·10−26 J T−1
1.660 538 782(83)·10−27 kg
9.109 382 15(45)·10−31 kg
7.297 352 5376(50)·10−3
1.380 6504(24)·10−23
Table C.1.: Fundamental physical constants relevant to the experiment (2006 CODATA
recommended values)
transition
4S1/2 ↔ 3D5/2
4S1/2 ↔ 3D3/2
4S1/2 ↔ 4P1/2
4S1/2 ↔ 4P3/2
3P1/2 ↔ 3D3/2
3P3/2 ↔ 3D3/2
3P3/2 ↔ 3D5/2
λair (nm)
411.042 129 776 393.3(1.0) THz
732.389
396.847
393.366
866.214
849.802
854.209
Isotope shift (MHz)
4134.711 720 (390)
4145(43)
706(42)
713(31)
-3464.3(3.0)
-3462.4(2.6)
-3465.4(3.7)
reference
[94, 122, 193]
[194]
[194]
[194]
[100]
[100]
[100]
Table C.2.: 40 Ca+ wavelength in air (λair ) and the corresponding isotope shifts (IS)
between 40 Ca+ and 43 Ca+ . The transition frequency for 4S1/2 ↔ 3D5/2 is given in THz.
quantity
life time 3D3/2
life time 3D5/2
life time 4P1/2
life time 4P3/2
gj (4S1/2 )
gj (3D5/2 )
value
1.20(1) s
1.168(7) s
7.098(20) ns
6.924(19) ns
2.00225664(9)
1.200 334 0(3)
Table C.3.: Atomic properties of calcium
109
reference
[96]
[96]
[97]
[97]
[195]
[122, 193]
Physical and optical properties of Calcium
m0
coefficient
-3/2
p
1/5
-1/2
p
2/5
p1/2
3/5
p3/2
4/5
5/2
1
Table C.4.: Clebsch Gordan Coefficients for the 40 Ca+ quadrupole transition S1/2 , m =
1/2 → D5/2 , m0 . The coupling strength for S1/2 , m = −1/2 → D5/2 , m0 can be derived by
exchanging m0j → −m0j .
-6
m´
-5
-2
-1
0
1
2
m = -2
0.03
0.06
0.11
0.17
0.24
m = -1
0.06
0.11
0.15
0.18
0.17
m=0
0.11
0.15
0.17
0.15
0.11
m=1
0.17
0.18
0.15
0.11
0.06
m=2
0.24
0.17
0.11
0.06
0.03
m = -2
0.09
0.17
0.24
0.29
0.31
0.28
m = -1
0.19
0.26
0.26
0.20
0.08
0.10
0.34
m=0
0.30
0.26
0.14
0.00
0.14
0.26
0.30
m=1
0.34
0.10
0.08
0.20
0.26
0.26
0.19
0.28
0.31
0.29
0.24
0.17
0.09
-4
32D5/2
-3
m=2
0.40
0.42
0.40
0.33
0.22
m = -1
0.41
0.39
0.27
0.09
0.09
0.27
0.39
0.41
m=0
0.48
0.12
0.14
0.29
0.34
0.29
0.14
0.12
0.48
0.41
0.39
0.27
0.09
0.09
0.27
0.39
0.41
0.22
0.33
0.40
0.42
0.40
0.33
0.22
0.48
0.57
0.57
0.52
0.44
0.34
0.23
0.12
m = -1
0.68
0.31
0.00
0.23
0.39
0.48
0.50
0.44
0.31
0.53
0.53
0.40
0.22
0.00
0.22
0.40
0.53
0.53
0.31
0.44
0.50
0.48
0.39
0.23
0.00
0.31
0.68
0.12
0.23
0.34
0.44
0.52
0.57
0.57
0.48
m=2
m=1
m=2
F´= 4
m = -2
m=1
1.00
6
F´= 3
0.33
m=0
m=0
5
F´= 2
0.22
m=2
m = -1
4
m = -2
m=1
m = -2
3
F´= 5
0.82
0.65
0.50
0.38
0.27
0.17
0.10
0.04
0.58
0.70
0.71
0.67
0.59
0.49
0.38
0.25
0.13
0.30
0.47
0.58
0.65
0.67
0.65
0.58
0.47
0.30
0.13
0.25
0.38
0.49
0.59
0.67
0.71
0.70
0.58
0.04
0.10
0.17
0.27
0.38
0.50
0.65
0.82
F´= 6
1.00
Figure C.1.: Coupling strength Λ(F, m, F 0 , m0 ) for the 43 Ca+ quadrupole transitions
S1/2 (F = 4, m) ↔ D5/2 (F 0 , m0 ) with ∆m = m0 −m neglecting the geometry and polarization
dependence. Figure taken from [94].
110
D. Journal Publications
The work described in this thesis gave rise to a number of journal publications which are:
1. ”High-fidelity entanglement of 43 Ca+ hyperfine clock states.”
G.Kirchmair, J.Benhelm, F.Zähringer, R.Gerritsma, C.F.Roos & R.Blatt
Phys. Rev. A, 79, 020304 (2009)
2. ”Deterministic entanglement of ions in thermal states of motion.”
G.Kirchmair, J.Benhelm, F.Zähringer, R.Gerritsma, C.F.Roos & R. Blatt
New J. Phys., 11, 023002 (2009)
3. ”State-independent experimental test of quantum contextuality”.
G. Kirchmair, F. Zähringer, R. Gerritsma, M. Kleinmann, O.Gühne, A.Cabello, R.Blatt &
C.F.Roos
Nature 460, 494 (2009).
4. ”Quantum simulation of the Dirac equation.”
R. Gerritsma, G.Kirchmair, F. Zähringer, E.Solano, R.Blatt & C.F.Roos
Nature 463, 68 (2010)
Further articles that have been published in the framework of this thesis:
5. ”Measurement of the hyperfine structure of the S1/2 -D5/2 transition in
J.Benhelm, G.Kirchmair, U. Rapol, T.Körber, C.F.Roos & R.Blatt,
Phys. Rev. A, 75, 032506 (2007)
43
Ca+ .”
6. ”Towards fault-tolerant quantum computing with trapped ions.”
J.Benhelm, G.Kirchmair, C.F.Roos & R.Blatt
Nat. Phys. 4, 463 (2008)
7. ”Experimental quantum-information processing with
J.Benhelm, G.Kirchmair, C.F.Roos & R.Blatt
43
Ca+ ions.”
Phys. Rev. A 77, 062306 (2008)
8. ”Precision measurement of the branching fractions of the 4 P3/2 decay of Ca II.”
R. Gerritsma, G. Kirchmair, F. Zähringer, J. Benhelm, R. Blatt, C. F. Roos Eur. Phys. J.
D 50,13 (2008)
9. ”Absolute Frequency Measurement of the 40 Ca+ 4s2 S1/2 − 3d2 D5/2 Clock Transition.”
M. Chwalla, J. Benhelm, K. Kim, G. Kirchmair, T. Monz, M. Riebe, P. Schindler, A. S. Villar,
W. Hänsel, C. F. Roos, R. Blatt, M. Abgrall, G. Santarelli, G. D. Rovera, Ph. Laurent
Phys. Rev. Lett. 102, 023002 (2009).
111
Journal Publications
10. ”Realization of a quantum walk with one and two trapped ions.”
F. Zähringer, G.Kirchmair, R. Gerritsma, E.Solano, R.Blatt & C.F.Roos
Phys. Rev. Lett. 104 100503 (2010)
11. ”Compatibility and noncontextuality for sequential measurements.”
O. Gühne, M. Kleinmann, A. Cabello, J.-A. Larsson, G. Kirchmair, F. Zähringer, R. Gerritsma, C.F. Roos
Phys. Rev. A 81, 022121 (2010)
112
Bibliography
[1] Benioff, P. Phys. Rev. Lett. 48, 1581 (1982).
[2] Feynman, R. Int. J. Theoret. Phys. 21, 467–488 (1982).
[3] Shor, P. W. In Proceedings of the 35th Annual Symposium on Foundations of Computer
Science, Santa Fe, NM, Nov. 20-22, IEEE Computer Society Press, pp. 124–134, (1994).
[4] Grover, L. K. Proceeding of the 28th ACM STOC , 212 (1996).
[5] Barenco, A., Bennett, C. H., Cleve, R., DiVincenzo, D. P., Margolus, N., Shor, P., Sleator,
T., Smolin, J. A., and Weinfurter, H. Phys. Rev. A 52(5), 3457–3467 (1995).
[6] DiVincenzo, D. P. Phys. Rev. A 51, 1015 (1995).
[7] DiVincenzo, D. P. Fortschr. Phys. 48, 771–783 (2000).
[8] Gershenfeld, N. A. and Chuang, I. L. Science 275(5298), 350–356 (1997).
[9] Makhlin, Y., Schön, G., and Shnirman, A. Rev. Mod. Phys. 73, 357 (2001).
[10] Majer, J., Chow, J. M., Gambetta, J. M., Koch, J., Johnson, B. R., Schreier, J. A., Frunzio,
L., Schuster, D. I., Houck, A. A., Wallraff, A., Blais, A., Devoret, M. H., Girvin, S. M., and
Schoelkopf, R. J. Nature 449, 443–447 (2007).
[11] Loss, D. and DiVincenzo, D. P. Phys. Rev. A 57, 120 (1998).
[12] Briegel, H. J., Calarco, T., Jaksch, D., Cirac, J., and Zoller, P. J. Mod. Opt 47, 415 (2000).
[13] Knill, E., Laflamme, R., and Milburn, G. J. Nature 409, 46–52 (2001).
[14] Cirac, J. I. and Zoller, P. Phys. Rev. Lett. 74(20), 4091–4094 (1995).
[15] Vandersypen, L. M. K. and Chuang, I. L. Rev. Mod. Phys. 76, 1037 (2004).
[16] Steffen, M., Ansmann, M., Bialczak, R. C., Katz, N., Lucero, E., McDermott, R., Neeley,
M., Weig, E. M., Cleland, A. N., and Martinis, J. M. Science 313, 1423–1425 (2006).
[17] Plantenberg, J. H., de Groot, P. C., Harmans, C. J. P. M., and Mooij, J. E. Nature 447,
836–839 (2007).
[18] DiCarlo, L., Chow, J. M., Gambetta, J. M., Bishop, L. S., Johnson, B. R., Schuster, D. I.,
Majer, J., Blais, A., Frunzio, L., and Schoelkopf1, S. M. G. . R. J. Nature 460, 240 (2009).
[19] Clarke, J. and Wilhelm, F. Nature 453, 1031 (2008).
113
Bibliography
[20] Karski, M., Förster, L., Choi, J.-M., Steffen, A., Alt, W., Meschede, D., and Widera, A.
Science 325, 174–177 (2009).
[21] Bloch, I. Nature 453, 1016–1022 (2008).
[22] Hofstetter, W., Cirac, J. I., Zoller, P., Demler, E., and Lukin, M. D. Phys. Rev. Lett. 89,
220407 (2002).
[23] Häffner, H., Roos, C., and Blatt, R. Physics Reports 469, 155 (2008).
[24] Wineland, D., Bergquist, J., Itano, W., and Drullinger, R. Opt. Lett. 5, 254 (1980).
[25] Nagourney, W., Sandberg, J., and Dehmelt, H. Phys. Rev. Lett. 56(26), 2797–2799 (1986).
[26] Sauter, T., Neuhauser, W., Blatt, R., and Toschek, P. E. Phys. Rev. Lett. 57(14), 1696–1698
(1986).
[27] Bergquist, J. C., Hulet, R. G., Itano, W. M., and Wineland, D. J. Phys. Rev. Lett. 57(14),
1699–1702 (1986).
[28] Bollinger, J. J., Heinzen, D. J., Itano, W. M., Gilbert, S. L., and Wineland, D. J. IEEE
Trans. Instr. Meas. 40, 126–128 (1991).
[29] Diedrich, F., Peik, E., Chen, J. M., Quint, W., and Walther, H. Phys. Rev. Lett. 59(26),
2931–2934 (1987).
[30] Wineland, D. J., Bergquist, J. C., Itano, W. M., Bollinger, J. J., and Manney, C. H.
Phys. Rev. Lett. 59(26), 2935–2938 (1987).
[31] Raizen, M. G., Gilligan, J. M., Bergquist, J. C., Itano, W. M., and Wineland, D. J.
Phys. Rev. A 45(9), 6493–6501 (1992).
[32] Kingdon, K. Phys. Rev. 21, 408 (1923).
[33] Monroe, C., Meekhof, D. M., King, B. E., Itano, W. M., and Wineland, D. J. Phys. Rev. Lett.
75(25), 4714–4717 (1995).
[34] Sackett, C. A., Kielpinski, D., King, B. E., Langer, C., Meyer, V., Myatt, C. J., Rowe, M.,
Turchette, Q. A., Itano, W. M., Wineland, D. J., and Monroe, C. Nature 404, 256–259
(2000).
[35] DeMarco, B., Ben-Kish, A., Leibfried, D., Meyer, V., Rowe, M., Jelenković, B. M., Itano,
W. M., Britton, J., Langer, C., Rosenband, T., and Wineland, D. J. Phys. Rev. Lett. 89(26),
267901 (2002).
[36] Schmidt-Kaler, F., Häffner, H., Riebe, M., Gulde, S., Lancaster, G. P. T., Deuschle, T.,
Becher, C., Roos, C. F., Eschner, J., and Blatt, R. Nature 422, 408–411 (2003).
[37] Leibfried, D., Blatt, R., Monroe, C., and Wineland, D. Rev. Mod. Phys. 75, 281 (2003).
[38] Haljan, P. C., Brickman, K.-A., Deslauriers, L., Lee, P. J., and Monroe, C. Phys. Rev. Lett.
94(15), 153602 (2005).
114
Bibliography
[39] Home, J. P., McDonnell, M. J., Lucas, D. M., Imreh, G., Keitch, B. C., Szwer, D. J., Thomas,
N. R., Webster, S. C., Stacey, D. N., and Steane, A. M. New J. Phys. 8, 188 (2006).
[40] Benhelm, J., Kirchmair, G., Roos, C. F., and Blatt, R. Nat. Phys. 4, 463–466 (2008).
[41] Riebe, M., Häffner, H., Roos, C. F., Hänsel, W., Benhelm, J., Lancaster, G. P. T., Körber,
T. W., Becher, C., Schmidt-Kaler, F., James, D. F. V., and Blatt, R. Nature 429, 734–737
(2004).
[42] Barrett, M. D., Chiaverini, J., Schaetz, T., Britton, J., Itano, W. M., Jost, J. D., Knill, E.,
Langer, C., Leibfried, D., Ozeri, R., and Wineland, D. J. Nature 429, 737–739 (2004).
[43] Chiaverini, J., Leibfried, D., Schaetz, T., Barrett, M. D., Blakestad, R. B., Britton, J., Itano,
W. M., Jost, J. D., Knill, E., Langer, C., Ozeri, R., and Wineland, D. J. Nature 432, 602–605
(2004).
[44] Reichle, R., Leibfried, D., Knill, E., Britton, J., Blakestad, R. B., Jost, J. D., Langer, C.,
Ozeri, R., Seidelin, S., and Wineland, D. J. Nature 443, 838–841 (2006).
[45] Leibfried, D., Knill, E., Seidelin, S., Britton, J., Blakestad, R. B., Chiaverini, J., Hume,
D. B., Itano, W. M., Jost, J. D., Langer, C., Ozeri, R., Reichle, R., and Wineland, D. J.
Nature 438(7068), 639–642 (2005).
[46] Häffner, H., Hänsel, W., Roos, C. F., Benhelm, J., Chek-al-Kar, D., Chwalla, M., Körber,
T., Rapol, U. D., Riebe, M., Schmidt, P. O., Becher, C., Gühne, O., Dür, W., and Blatt, R.
Nature 438, 643–646 (2005).
[47] Gulde, S. Experimental realization of quantum gates and the Deutsch-Josza algorithm with
trapped 40 Ca+ –ions. PhD thesis, Universität Innsbruck, (2003).
[48] Brickman, K.-A., Haljan, P. C., Lee, P. J., Acton, M., Deslauriers, L., and Monroe, C.
Phys. Rev. A 72, 050306–1 (2005).
[49] Roos, C. F., Lancaster, G. P. T., Riebe, M., Häffner, H., Hänsel, W., Gulde, S., Becher, C.,
Eschner, J., Schmidt-Kaler, F., and Blatt, R. Phys. Rev. Lett. 92, 220402 (2004).
[50] Riebe, M., Kim, K., Schindler, P., Monz, T., Schmidt, P. O., Körber, T. K., Hänsel, W.,
Häffner, H., Roos, C. F., and Blatt, R. Phys. Rev. Lett. 97, 220407 (2006).
[51] Kielpinski, D., Meyer, V., Rowe, M. A., Sackett, C. A., Itano, W. M., Monroe, C., and
Wineland, D. J. Science 291(5506), 1013–1015 (2001).
[52] Monz, T., Kim, K., Villar, A. S., Schindler, P., Chwalla, M., Riebe, M., Roos, C. F., Häffner,
H., Hänsel, W., Hennrich, M., and Blatt, R. Phys. Rev. Lett. 103, 200503 (2009).
[53] Moehring, D. L., Maunz, P., Olmschenk, S., Younge, K. C., Matsukevich, D. N., Duan,
L.-M., and Monroe, C. Nature 449, 68–71 (2007).
[54] Olmschenk, S., Matsukevich, D. N., Maunz, Hayes, Duan, M., and Monroe, C. Science 323,
486 (2009).
115
Bibliography
[55] Home, J. P., Hanneke, D., Jost, J. D., Amini, J. M., Leibfried, D., and Wineland, D. J.
Science 325, 1227–1230 (2009).
[56] Hanneke, D., Home, J. P., Jost, J. D., Amini, J. M., Leibfried, D., and Wineland, D. J. Nat.
Phys. 6, 13 (2009).
[57] Shor, P. W. Phys. Rev. A 52(4), R2493–R2496 (1995).
[58] Steane, A. M. Phys. Rev. Lett. 77(5), 793–797 (1996).
[59] Shor, P. 37th Annual Symposium on Foundations of Computer Science (FOCS ’96) 00, 56
(1996).
[60] Knill, E. Nature 434, 39–44 (2005).
[61] Raussendorf, R. and Harrington, J. Phys. Rev. Lett. 98, 190504 (2007).
[62] Reichardt, B. W. arXiv:quant-ph/0406025 (2004).
[63] Friedenauer, H., Schmitz, H., Glueckert, J., Porras, D., and Schaetz, T. Nat. Phys. 4, 757–761
(2008).
[64] Einstein, A., Podolsky, B., and Rosen, N. Phys. Rev. 47, 777–780 (1935).
[65] Bell, J. Physics 1, 195 (1965).
[66] Weihs, G., Jennewein, T., Simon, C., Weinfurter, H., and Zeilinger, A. Phys. Rev. Lett. 81,
5039–5043 (1998).
[67] Aspect, A., Grangier, P., and Roger, G. Phys. Rev. Lett. 49, 91–94 (1982).
[68] Tittel, W., Brendel, J., Zbinden, H., and Gisin, N. Phys. Rev. Lett. 81, 3563–3566 (1998).
[69] Rowe, M. A., Kielpinski, D., Meyer, V., Sackett, C. A., Itano, W. M., Monroe, C., and
Wineland, D. J. Nature 409, 791–794 (2001).
[70] Specker, E. Dialectica 14, 239–246 (1960).
[71] Kochen, S. and Specker, E. P. J. Math. Mech. 17, 59––87 (1967).
[72] Bell, J. S. Rev. Mod. Phys. 38, 447—-452 (1966).
[73] Amselem, E., Rådmark, M., Bourennane, M., and Cabello, A. Phys. Rev. Lett. 103, 160405
(2009).
[74] Moussa, O., Ryan, C. A., Cory, D. G., and Laflamme, R. arXiv:0912.0485 (2009).
[75] Kirchmair, G., Zähringer, F., Gerritsma, R., Kleinmann, M., O., G., Cabello, A., Blatt, R.,
and Roos, C. F. Nature 460, 494 (2009).
[76] Kirchmair, G., Benhelm, J., Zähringer, F., Gerritsma, R., Roos, C. F., and Blatt, R. New
J. Phys. 11, 023002 (2009).
116
Bibliography
[77] Kirchmair, G., Benhelm, J., Zähringer, F., Gerritsma, R., Roos, C. F., and Blatt, R.
Phys. Rev. A 79, 020304(R) (2009).
[78] Gerritsma, R., Kirchmair, G., Zähringer, F., Solano, E., Blatt, R., and Roos, C. F. Nature
463, 68 (2010).
[79] Nielsen, M. A. and Chuang, I. L. Quantum computation and quantum information. Cambridge Univ. Press, Cambridge, (2000).
[80] Sørensen, A. and Mølmer, K. Phys. Rev. Lett. 82, 1971–1974 (1999).
[81] Sørensen, A. and Mølmer, K. Phys. Rev. A 62, 022311 (2000).
[82] Clauser, J. and Shimony, A. Rep. Prog. Phys. 41, 1883 (1978).
[83] Everett, H. Rev. Mod. Phys. 29, 454 (1957).
[84] Gröblacher, S., Paterek, T., Kaltenbaek, R., Brukner, S., Zdotukowski, M., Aspelmeyer, M.,
and Zeilinger, A. Nature 446, 871 (2007).
[85] Rauschenbeutel, A., Nogues, G., Osnaghi, S., Bertet, P., Brune, M., Raimond, J. M., and
Haroche, S. Science 288(5473), 2024–2028 (2000).
[86] Aspelmeyer, M., Böm, H. R., Gyatso, T., Jennewein, T., Kaltenbaek, R., Lindenthal, M.,
Molina-Terriza, G., Poppe, A., Resch, K., Taraba, M., Ursin, R., Walther, P., and Zeilinger,
A. Science 301, 621–623 (2003).
[87] Mandel, O., Greiner, M., Widera, A., Rom, T., Hänsch, T. W., and Bloch, I. Phys. Rev. Lett.
91, 010407 (2003).
[88] Jennewein, T., Simon, C., Weihs, G., Weinfurter, H., and Zeilinger, A. Phys. Rev. Lett.
84(20), 4729–4732 (2000).
[89] Raussendorf R., Browne D., B. l. J. Phys. Rev. A 68, 022312 (2003).
[90] Cohen-Tannoudji, Bernhard Diu, F. L. Quantum mechanics I + II. Wiley, (1977).
[91] Haroche, S. and Raimond, J.-M. Exploring the quantum. Oxford Univ. Press, Oxford, (2006).
[92] Roos, C. Controlling the quantum state of trapped ions. PhD thesis, Universität Innsbruck,
(2000).
[93] Riebe, M. Preparation of entangled states and quantum teleportation with atomic qubits.
PhD thesis, Universität Innsbruck, (2005).
[94] Benhelm, J. Precision spectroscopy and quantum information processsing with trapped calcium ions. PhD thesis, Universität Innsbruck, (2008).
[95] Gerritsma, R., Kirchmair, G., Zähringer, F., Benhelm, J., Blatt, R., and Roos, C.
Eur. Phys. J. D 50 (2008).
117
Bibliography
[96] Barton, P. A., Donald, C. J. S., Lucas, D. M., Stevens, D. A., Steane, A. M., and Stacey,
D. N. Phys. Rev. A 62, 032503 (2000).
[97] Jin, J. and Church, D. A. Phys. Rev. Lett. 70, 3213–3216 (1993).
[98] James, D. F. V. Appl. Phys. B 66, 181–190 (1998).
[99] Arbes, F., Benzing, M., Gudjons, T., Kurth, F., and Werth, G. Z. Phys. D 31, 27–30 (1994).
[100] Nörtershauser, W., Blaum, K., Icker, K., Muller, P., Schmitt, A., Wendt, K., and Wiche, B.
Eur. Phys. J. D , 33–39 (1998).
[101] Breit, G. and Rabi, I. I. Phys Rev 38, 2082 (1931).
[102] Benhelm, J., Kirchmair, G., Roos, C. F., and Blatt, R. Phys. Rev. A 77, 062306 (2008).
[103] Häffner, H., Gulde, S., Riebe, M., Lancaster, G., Becher, C., Eschner, J., Schmidt-Kaler, F.,
and Blatt, R. Phys. Rev. Lett. 90, 143602 (2003).
[104] Loudon, R. The Quantum Theory of Light. Oxford Univ. Press, (1973).
[105] Schindler, P. Master’s thesis, Universität Innsbruck, (2007).
[106] Johanning, M., Braun, A., Timoney, N., Elman, V., Neuhauser, W., and Wunderlich, C.
Phys. Rev. Lett. 102, 073004 (2009).
[107] Ospelkaus, C., Langer, C. E., Amini, J. M., Brown, K. R., Leibfried, D., and Wineland, D. J.
Phys. Rev. Lett. 101, 090502 (2008).
[108] Lee, P. J., Brickman, K. A., Deslauriers, L., Haljan, P. C., Duan, L., and Monroe, C. J. Opt. B
7, 371 (2005).
[109] Leibfried, D., DeMarco, B., Meyer, V., Lucas, D., Barrett, M., Britton, J., Itano, W. M.,
Jelenković, B., Langer, C., Rosenband, T., and Wineland, D. J. Nature 422, 412–415 (2003).
[110] Milburn, G. J., Schneider, S., and James, D. F. Fortschr. Phys. 48, 801–810 (2000).
[111] Solano, E., de Matos Filho, R. L., and Zagury, N. Phys. Rev. A 59, R2539–R2543 (1999).
[112] Roos, C. F., Monz, T., Kim, K., Riebe, M., Häffner, H., James, D. F. V., and Blatt, R.
Phys. Rev. A 77, 040302(R) (2008).
[113] Schmidt-Kaler, F., Häffner, H., Gulde, S., Riebe, M., Lancaster, G. P. T., Deuschle, T.,
Becher, C., Hänsel, W., Eschner, J., Roos, C. F., and Blatt, R. Appl. Phys. B 77, 789
(2003).
[114] Unanyan, R. G. and Fleischhauer, M. Phys. Rev. Lett. 90, 133601 (2003).
[115] Lougovski, P., Walther, H., and Solano, E. Eur. Phys. J. D 38, 423–426 (2006).
[116] Santos, M. F., Giedke, G., and Solano, E. Phys. Rev. Lett. 98, 020401 (2007).
[117] Lamata, L., León, J., Schätz, T., and Solano, E. Phys. Rev. Lett. 98, 253005 (2007).
118
Bibliography
[118] Kirchmair, G. Master’s thesis, Universität Innsbruck, (2007).
[119] Notcutt, M., Ma, L.-S., Ye, J., and Hall, J. L. Opt. Lett. 30, 1815–1817 (2005).
[120] Drever, R. W. P., Hall, J. L., Kowalski, F. V., Hough, J., Ford, G. M., Munley, A. J., and
Ward, H. Appl. Phys. B 31, 97–105 (1983).
[121] Ma, L., Jungner, P., Ye, J., and Hall, J. L. Opt. Lett. 19, 1777 (1994).
[122] Chwalla, M. Precision spectroscopy with 40 Ca+ ions in a Paul trap. PhD thesis, Institute
for Experimental Physics, Innsbruck, (2009).
[123] Pham, P. T. T. Master’s thesis, Massachusetts Institute of Technology, (2005).
[124] Rotter, D. Master’s thesis, University of Innsbruck, (2003).
[125] Ramsey, N. F. Phys. Rev. 78, 695–699 (1950).
[126] Benhelm, J., Kirchmair, G., Rapol, U., Körber, T., Roos, C. F., and Blatt, R. Phys. Rev. A
75, 032506 (2007).
[127] Kielpinski, D., Monroe, C., and Wineland, D. J. Nature 417, 709–711 (2002).
[128] Hensinger, W. K., Olmschenk, S., Stick, D., Hucul, D., Yeo, M., Acton, M., Deslauriers, L.,
and Monroe, C. Appl. Phys. Lett. 88, 034101 (2006).
[129] Blakestad, R. B., Ospelkaus, C., VanDevender, A. P., Amini, J. M., Britton, J., Leibfried,
D., and Wineland, D. J. Phys. Rev. Lett. 102, 153002 (2009).
[130] Splatt, F., Harlander, M., Brownnutt, M., Zähringer, F., Blatt, R., and Hänsel, W. New
J. Phys. 11, 103008 (2009).
[131] Home, J. P. and Steane, A. M. Quant. Inf. Comp. 6, 289 (2006).
[132] Turchette, Q. A., Kielpinski, King, B. E., Leibfried, D., Meekhof, D. M., Myatt, C. J.,
Rowe, M. A., Sackett, C. A., Wood, C. S., Itano, W. M., Monroe, C., and Wineland, D. J.
Phys. Rev. A 61, 063418 (2000).
[133] Deslauriers, L., Haljan, P. C., Lee, P. J., Brickman, K.-A., Blinov, B. B., Madsen, M. J., and
Monroe, C. Phys. Rev. A 70, 043408 (2004).
[134] Labaziewicz, J., Ge, Y., Antohi, P., Leibrandt, D., Brown, K., and Chuang, I. Phys. Rev. Lett.
100, 013001 (2008).
[135] Nebendahl, V., Häffner, H., and Roos, C. F. Phys. Rev. A 79, 012312 (2009).
[136] Hradil, Z., Řeháček, J., Fiurášek, J., and Ježek, M. Lect. Notes Phys. 649, 59–112 (2004).
[137] Jez̆ek, M., Fiurás̆ek, J., and Hradil, Z. Phys. Rev. A 68, 012305 (2003).
[138] Steane, A. Rep. Prog. Phys. 61, 117–173 (1998).
119
Bibliography
[139] Rowe, M. A., Ben-Kish, A., DeMarco, B., Leibfried, D., Meyer, V., Beall, J., Britton, J.,
Hughes, J., Itano, W. M., Jelenković, B., Langer, C., Rosenband, T., and Wineland, D. J.
Quant. Inf. Comp. 2, 257–271 (2002).
[140] Blinov, B., Deslauriers, L., Lee, P., Madsen, M., Miller, R., and Monroe, C. Phys. Rev. A
65, 040304 (2002).
[141] Barrett, M. D., DeMarco, B., Schaetz, T., Leibfried, D., Britton, J., Chiaverini, J., Itano,
W. M., Jelenković, B., Jost, J. D., Langer, C., Rosenband, T., and Wineland, D. J.
Phys. Rev. A 68, 042302 (2003).
[142] Harris, F. J. Proc. IEEE 66, 51–83 (1978).
[143] Leibfried, D., Barrett, M. D., Schaetz, T., Britton, J., Chiaverini, J., Itano, W. M., Jost,
J. D., Langer, C., and Wineland, D. J. Science 304, 1476–1478 (2004).
[144] Haljan, P. C., Lee, P. J., Brickman, K.-A., Acton, M., Deslauriers, L., and Monroe, C.
Phys. Rev. A 72, 062316 (2005).
[145] Leibfried, D., Knill, E., Ospelkaus, C., and Wineland, D. J. Phys. Rev. A 76, 032324 (2007).
[146] Roos, C. F. New J. Phys. 10, 013002 (2008).
[147] Langer, C., Ozeri, R., Jost, J. D., Chiaverini, J., DeMarco, B., Ben-Kish, A., Blakestad,
R. B., Britton, J., Hume, D. B., Itano, W. M., Leibfried, D., Reichle, R., Rosenband, T.,
Schaetz, T., Schmidt, P. O., and Wineland, D. J. Phys. Rev. Lett. 95, 060502 (2005).
[148] Olmschenk, S., Younge, K. C., Moehring, D. L., Matsukevich, D. N., Maunz, P., and Monroe,
C. Phys. Rev. A 76, 052314 (2007).
[149] Kim, K., Chang, M., Islam, R., Korenblit, S., Duan, L., and Monroe, C. Phys. Rev. Lett.
103, 120502 (2009).
[150] Nebendahl, V. Master’s thesis, University of Innsbruck, (2008).
[151] Wang, X., Sørensen, A., and Mølmer, K. Phys. Rev. Lett. 86(17), 3907–3910 (2001).
[152] Myerson, A. H., Szwer, D. J., Webster, S. C., Allcock, D. T. C., Curtis, M. J., Imreh, G.,
Sherman, J. A., Stacey, D. N., Steane, A. M., and Lucas, D. M. Phys. Rev. Lett. 100, 200502
(2008).
[153] Knill, E., Leibfried, D., Reichle, R., Britton, J., Blakestad, R. B., Jost, J. D., Langer, C.,
Ozeri, R., Seidelin, S., and Wineland, D. J. Phys. Rev. A 77, 012307 (2008).
[154] Benhelm, J., Kirchmair, G., Roos, C. F., and Blatt, R. arXiv:0804.1261 (2008).
[155] Lucas, D. M., Keitch, B. C., Home, J. P., Imreh, G., McDonnell, M. J., Stacey, D. N., Szwer,
D. J., and Steane, A. M. arXiv: 0710.4421v1 (2007).
[156] Bohr, N. Phys. Rev. 48, 696–702 (1935).
120
Bibliography
[157] Cabello, A. and Garcı́a-Alcaine, G. Phys. Rev. Lett. 80, 1797–1799 (1998).
[158] Meyer, D. A. Phys. Rev. Lett. 83, 3751–1754 (1999).
[159] Cabello, A. Phys. Rev. Lett. 101, 210401 (2008).
[160] Gühne, O., Kleinmann, M., Cabello, A., Larsson, J., Kirchmair, G., Zähringer, F., Gerritsma,
R., and Roos, C. arXiv:0912.4846 (2010).
[161] Peres, A. Phys. Lett. A 151, 107–108 (1990).
[162] Mermin, N. D. Phys. Rev. Lett. 65, 3373—-3376 (1990).
[163] Garay, L. J., Anglin, J. R., Cirac, J. I., and Zoller, P. Phys. Rev. Lett. 85, 4643–4647 (2000).
[164] Schliemann, J., Loss, D., and Westervelt, R. M. Phys. Rev. Lett. 94, 206801 (2005).
[165] Cserti, J. and Dávid, G. Phys. Rev. B 74, 172305 (2006).
[166] Katsnelson, M. I., Novoselov, K. S., and Geim, A. K. Nat. Phys. 2, 620–625 (2006).
[167] Neto, A. H. C., Guinea, F., Peres, N. M. R., Novoselov, K. S., and Geim, A. K.
Rev. Mod. Phys. 81, 109–162 (2009).
[168] Thaller, B. The Dirac equation. Springer-Verlag, (1992).
[169] Anderson, C. D. Phys. Rev. 43, 491–494 (1933).
[170] Klein, O. Z. Phys. 53, 157–165 (1929).
[171] Schrödinger, E. Sitz. Preuss. Akad. Wiss. Phys.-Math. Kl. 24, 418–428 (1930).
[172] Bermudez, A., Martin-Delgado, M. A., and Solano, E. Phys. Rev. A 76, 041801(R) (2007).
[173] Zhang, X. Phys. Rev. Lett. 100, 113903 (2008).
[174] Vaishnav, J. Y. and Clark, C. W. Phys. Rev. Lett. 100, 153002 (2008).
[175] Otterbach, J., Unanyan, R. G., and Fleischhauer, M. Phys. Rev. Lett. 102, 063602 (2009).
[176] Leibfried, D., DeMarco, B., Meyer, V., Rowe, M., Ben-Kish, A., Britton, J., Itano, W. M.,
Jelenković, B., Langer, C., Rosenband, T., and Wineland, D. J. Phys. Rev. Lett. 89, 247901
(2002).
[177] Porras, D. and Cirac, J. I. Phys. Rev. Lett. 92, 207901 (2004).
[178] Johanning, M., Varón, A. F., and Wunderlich, C. J. Phys. B 42, 154009 (2009).
[179] Thaller, B. arXiv:quant-ph/0409079 (2004).
[180] Schwabl, F. Quantenmechanik für Fortgeschrittene. Springer, (2005).
[181] Krekora, P., Su, Q., and Grobe, R. Phys. Rev. Lett. 93, 043004 (2004).
121
Bibliography
[182] Wang, Z.-Y. and Xiong, C.-D. Phys. Rev. A 77, 045402 (2008).
[183] Wallentowitz, S. and Vogel, W. Phys. Rev. Lett. 75, 2932–2935 (1995).
[184] Meekhof, D. M., Monroe, C., King, B. E., Itano, W. M., and Wineland, D. J. Phys. Rev. Lett.
76(11), 1796–1799 (1996).
[185] Garcı́a-Ripoll, J. J., Zoller, P., and Cirac, J. I. Phys. Rev. Lett. 91(15), 157901 (2003).
[186] Zähringer, F., Kirchmair, G., Gerritsma, R., Solano, E., Blatt, R., and Roos, C. F.
arXiv:0911.1876 (2009).
[187] Schmitz, H., Matjeschk, R., Schneider, C., Glueckert, J., Enderlein, M., Huber, T., and
Schaetz, T. Phys. Rev. Lett. 103, 090504 (2009).
[188] Menicucci, N. C. and Milburn, G. J. Phys. Rev. A 76, 052105 (2007).
[189] Alsing, P. M., Dowling, J. P., and Milburn, G. J. Phys. Rev. Lett. 94, 220401 (2005).
[190] Porras, D., Marquardt, F., von Delft, J., and Cirac, J. I. Phys. Rev. A 78, 010101 (2008).
[191] Schützhold, R., Uhlmann, M., Petersen, L., Schmitz, H., Friedenauer, A., and Schätz, T.
Phys. Rev. Lett. 99, 201301 (2007).
[192] Garcı́a-Mata, I., Zhirov, O. V., and Shepelyansky, D. L. Eur. Phys. J. D 41, 325–330 (2007).
[193] Chwalla, M., Benhelm, J., Kim, K., Kirchmair, G., Monz, T., Riebe, M., Schindler, P., Villar,
A. S., Hänsel, W., and Roos, C. F. Phys. Rev. Lett. 102, 023002 (2009).
[194] Kurth, F., Gudjons, T., Hilbert, B., Reisinger, T., Werth, G., and Martensson-Pendrill,
A. M. Z. Phys. D 34, 227–232 (1995).
[195] Tommaseo, G., Pfeil, T., Revalde, G., Werth, G., Indelicato, P., and Desclaux, J.
Eur. Phys. J. D 25, 113–121 (2003).
122
Danksagung
Das Gelingen dieser Arbeit wäre ohne die Hilfe und Geduld Vieler nicht möglich gewesen. Ihnen
allen möchte hiermit herzlich danken.
An erster Stelle möchte ich mich bei Rainer Blatt bedanken, der es mir ermöglicht hat meine
Diplomarbeit und jetzt auch meine Doktorarbeit in seiner Arbeitsgruppe anzufertigen. Durch
sein stetiges Bemühen und seinen großen Einsatz hat er ein Klima und wissenschaftliches Umfeld
geschaffen indem es eine Freude ist zu arbeiten.
Ein großer Dank gebührt natürlich Christian Roos. Viele der in dieser Arbeit vorgestellten
Experimente waren nur durch seine Ideen und Unterstützung möglich. Besonders Bedanken möchte
ich mich auch für das Korrekturlesen meiner Arbeit.
Meine Arbeitsweise im Labor wurde am meisten durch Jan Benhelm geprägt. In den vier Jahren
unserer Zusammenarbeit konnte ich sehr viel von ihm lernen und meine Doktorarbeit profitierte
sehr von seiner sorgfältigen Aufbauarbeit.
Weiters Bedanken möchte ich mich bei Rene Gerritsma und Florian Zähringer die mit mir
unzählige Tage im Labor verbracht haben. Die interessanten und witzigen Unterhaltungen während
der langen Stunden vor dem Laborrechner werden mir immer in guter Erinnerung bleiben.
Vielen Dank auch an die ganze 40 Ca+ Crew für die gute Zusammenarbeit in den letzten Jahren.
Besonders bedanken möchte ich mich bei Philipp Schindler für das betreuen unserer Pulse Box
und Michael Chwalla für die Arbeit an den 729 nm cavities und der ”real laser front”.
Ein großer Dank gebührt auch der ganzen Kochgruppe die, jeden Tag, für ein sehr gutes Essen
und eine gute Stimmung während der Mittagspause gesorgt hat.
Nicht zu vergessen sind natürlich meine Freunde. Einen Physiker von seiner Arbeit abzulenken
ist nicht immer ganz einfach deshalb ist es ihnen zu verdanken, dass meine sozialen Kompetenzen
nicht auf der Strecke geblieben sind.
Ohne die Unterstützung meiner Familie hätte ich diese Doktorarbeit wohl nicht so erfolgreich
abschliessen können. Deshalb will ich mich an dieser Stelle auch recht herzlich bei ihnen bedanken.
Ein besonderer Dank gebührt meiner Freundin Astrid die mich all die Jahre durch mein Studium
und meine Doktorarbeit begleitet hat. Ihr Verständnis und ihre Unterstützung haben mir geholfen
auch die schwierigsten Situationen zu meistern.
123

Quantum non-demolition - Quantum Optics and Spectroscopy

Transcription

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